Displaying 1-10 of 38 results found.
Number of partitions of n into an odd number of parts.
+10
206
0, 1, 1, 2, 2, 4, 5, 8, 10, 16, 20, 29, 37, 52, 66, 90, 113, 151, 190, 248, 310, 400, 497, 632, 782, 985, 1212, 1512, 1851, 2291, 2793, 3431, 4163, 5084, 6142, 7456, 8972, 10836, 12989, 15613, 18646, 22316, 26561, 31659, 37556, 44601, 52743, 62416, 73593, 86809, 102064, 120025, 140736
COMMENTS
Number of partitions of n in which greatest part is odd.
Number of partitions of n+1 into an even number of parts, the least being 1. Example: a(5)=4 because we have [5,1], [3,1,1,1], [2,1,1] and [1,1,1,1,1,1].
Also number of partitions of n+1 such that the largest part is even and occurs only once. Example: a(5)=4 because we have [6], [4,2], [4,1,1] and [2,1,1,1,1]. - Emeric Deutsch, Apr 05 2006 Also the number of partitions of n such that the number of odd parts and the number of even parts have opposite parities. Example: a(8)=10 is a count of these partitions: 8, 611, 521, 431, 422, 41111, 332, 32111, 22211, 2111111. - Clark Kimberling, Feb 01 2014, corrected Jan 06 2021 In Chaves 2011 see page 38 equation (3.20). - Michael Somos, Dec 28 2014 Suppose that c(0) = 1, that c(1), c(2), ... are indeterminates, that d(0) = 1, and that d(n) = -c(n) - c(n-1)*d(1) - ... - c(0)*d(n-1). When d(n) is expanded as a polynomial in c(1), c(2),..,c(n), the terms are of the form H*c(i_1)*c(i_2)*...*c(i_k). Let P(n) = [c(i_1), c(i_2), ..., c(i_k)], a partition of n. Then H is negative if P has an odd number of parts, and H is positive if P has an even number of parts. That is, d(n) has A027193(n) negative coefficients, A027187(n) positive coefficients, and A000041 terms. The maximal coefficient in d(n), in absolute value, is A102462(n). - Clark Kimberling, Dec 15 2016
REFERENCES
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 39, Example 7.
FORMULA
G.f.: Sum(k>=1, x^(2*k-1)/Product(j=1..2*k-1, 1-x^j ) ). - Emeric Deutsch, Apr 05 2006 G.f.: - Sum(k>=1, (-x)^(k^2)) / Product(k>=1, 1-x^k ). - Joerg Arndt, Feb 02 2014 G.f.: Sum(k>=1, x^(k*(2*k-1)) / Product(j=1..2*k, 1-x^j)). - Michael Somos, Dec 28 2014
EXAMPLE
G.f. = x + x^2 + 2*x^3 + 2*x^4 + 4*x^5 + 5*x^6 + 8*x^7 + 10*x^8 + 16*x^9 + 20*x^10 + ...
The a(1) = 1 through a(8) = 10 partitions into an odd number of parts are the following. The Heinz numbers of these partitions are given by A026424. (1) (2) (3) (4) (5) (6) (7) (8)
(111) (211) (221) (222) (322) (332)
(311) (321) (331) (422)
(11111) (411) (421) (431)
(21111) (511) (521)
(22111) (611)
(31111) (22211)
(1111111) (32111)
(41111)
(2111111)
The a(1) = 1 through a(8) = 10 partitions whose greatest part is odd are the following. The Heinz numbers of these partitions are given by A244991. (1) (11) (3) (31) (5) (33) (7) (53)
(111) (1111) (32) (51) (52) (71)
(311) (321) (322) (332)
(11111) (3111) (331) (521)
(111111) (511) (3221)
(3211) (3311)
(31111) (5111)
(1111111) (32111)
(311111)
(11111111)
(End)
MAPLE
g:=sum(x^(2*k)/product(1-x^j, j=1..2*k-1), k=1..40): gser:=series(g, x=0, 50): seq(coeff(gser, x, n), n=1..45); # Emeric Deutsch, Apr 05 2006
MATHEMATICA
nn=40; CoefficientList[Series[ Sum[x^(2j+1)Product[1/(1- x^i), {i, 1, 2j+1}], {j, 0, nn}], {x, 0, nn}], x] (* Geoffrey Critzer, Dec 01 2012 *) a[ n_] := If[ n < 0, 0, Length@Select[ IntegerPartitions[ n], OddQ[ Length@#] &]]; (* Michael Somos, Dec 28 2014 *) a[ n_] := If[ n < 1, 0, Length@Select[ IntegerPartitions[ n], OddQ[ First@#] &]]; (* Michael Somos, Dec 28 2014 *) a[ n_] := If[ n < 0, 0, Length@Select[ IntegerPartitions[ n + 1], #[[-1]] == 1 && EvenQ[ Length@#] &]]; (* Michael Somos, Dec 28 2014 *) a[ n_] := If[ n < 1, 0, Length@Select[ IntegerPartitions[ n + 1], EvenQ[ First@#] && (Length[#] < 2 || #[[1]] != #[[2]]) &]]; (* Michael Somos, Dec 28 2014 *)
PROG
(PARI) {a(n) = if( n<1, 0, polcoeff( sum( k=1, n, if( k%2, x^k / prod( j=1, k, 1 - x^j, 1 + x * O(x^(n-k)) ))), n))}; /* Michael Somos, Jul 24 2012 */ (PARI) q='q+O('q^66); concat([0], Vec( (1/eta(q)-eta(q)/eta(q^2))/2 ) ) \\ Joerg Arndt, Mar 23 2014
CROSSREFS
The even-length version is A027187. The case of odd maximum as well as length is A340385. Other cases of odd length:
- A024429 counts set partitions of odd length. - A067659 counts strict partitions of odd length. - A089677 counts ordered set partitions of odd length. - A166444 counts compositions of odd length. - A174726 counts ordered factorizations of odd length. - A332304 counts strict compositions of odd length. - A339890 counts factorizations of odd length. A026804 counts partitions whose least part is odd.
A072233 counts partitions by sum and length.
A101707 counts partitions of odd positive rank.
Number of partitions of n into distinct parts such that number of parts is odd.
+10
85
0, 1, 1, 1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 9, 11, 14, 16, 19, 23, 27, 32, 38, 44, 52, 61, 71, 82, 96, 111, 128, 148, 170, 195, 224, 256, 293, 334, 380, 432, 491, 557, 630, 713, 805, 908, 1024, 1152, 1295, 1455, 1632, 1829, 2048, 2291, 2560, 2859, 3189, 3554, 3958, 4404
COMMENTS
Ramanujan theta functions: phi(q) := Sum_{k=-oo..oo} q^(k^2) ( A000122), chi(q) := Prod_{k>=0} (1+q^(2k+1)) ( A000700).
FORMULA
Expansion of (1-phi(-q))/(2*chi(-q)) in powers of q where phi(),chi() are Ramanujan theta functions. - Michael Somos, Feb 14 2006 G.f.: sum(n>=1, q^(2*n^2-n) / prod(k=1..2*n-1, 1-q^k ) ). [ Joerg Arndt, Apr 01 2014]
EXAMPLE
The a(5) = 1 through a(15) = 14 partitions (A-F = 10..15):
5 6 7 8 9 A B C D E F
321 421 431 432 532 542 543 643 653 654
521 531 541 632 642 652 743 753
621 631 641 651 742 752 762
721 731 732 751 761 843
821 741 832 842 852
831 841 851 861
921 931 932 942
A21 941 951
A31 A32
B21 A41
B31
C21
54321
(End)
MAPLE
b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2, 0,
`if`(n=0, t, add(b(n-i*j, i-1, abs(t-j)), j=0..min(n/i, 1))))
end:
a:= n-> b(n$2, 0):
MATHEMATICA
b[n_, i_, t_] := b[n, i, t] = If[n > i*(i + 1)/2, 0, If[n == 0, t, Sum[b[n - i*j, i - 1, Abs[t - j]], {j, 0, Min[n/i, 1]}]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Jan 16 2015, after Alois P. Heinz *) CoefficientList[Normal[Series[(QPochhammer[-x, x]-QPochhammer[x])/2, {x, 0, 100}]], x] (* Andrey Zabolotskiy, Apr 12 2017 *) Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&OddQ[Length[#]]&]], {n, 0, 30}] (* Gus Wiseman, Jan 09 2021 *)
PROG
(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( (eta(x^2+A)/eta(x+A) - eta(x+A))/2, n))} /* Michael Somos, Feb 14 2006 */ (PARI) N=66; q='q+O('q^N); S=1+2*sqrtint(N);
gf=sum(n=1, S, (n%2!=0) * q^(n*(n+1)/2) / prod(k=1, n, 1-q^k ) );
(PARI) N=66; q='q+O('q^N); S=1+sqrtint(N);
gf=sum(n=1, S, q^(2*n^2-n) / prod(k=1, 2*n-1, 1-q^k ) );
CROSSREFS
Numbers with these strict partitions as binary indices are A000069. The Heinz numbers of these partitions are A030059. Other cases of odd length:
- A024429 counts set partitions of odd length. - A089677 counts ordered set partitions of odd length. - A174726 counts ordered factorizations of odd length. - A339890 counts factorizations of odd length. A008289 counts strict partitions by sum and length.
A026804 counts partitions whose least part is odd, with strict case A026832.
Number of odd-length factorizations of n into factors > 1.
+10
78
0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 2, 1, 4, 1, 1, 1, 4, 1, 1, 1, 3, 1, 2, 1, 2, 2, 1, 1, 6, 1, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 5, 1, 1, 2, 5, 1, 2, 1, 2, 1, 2, 1, 8, 1, 1, 2, 2, 1, 2, 1, 6, 2, 1, 1, 5, 1, 1, 1
EXAMPLE
The a(n) factorizations for n = 24, 48, 60, 72, 96, 120:
24 48 60 72 96 120
2*2*6 2*3*8 2*5*6 2*4*9 2*6*8 3*5*8
2*3*4 2*4*6 3*4*5 2*6*6 3*4*8 4*5*6
3*4*4 2*2*15 3*3*8 4*4*6 2*2*30
2*2*12 2*3*10 3*4*6 2*2*24 2*3*20
2*2*2*2*3 2*2*18 2*3*16 2*4*15
2*3*12 2*4*12 2*5*12
2*2*2*3*3 2*2*2*2*6 2*6*10
2*2*2*3*4 3*4*10
2*2*2*3*5
MAPLE
g:= proc(n, k, t) option remember; `if`(n>k, 0, t)+
`if`(isprime(n), 0, add(`if`(d>k, 0, g(n/d, d, 1-t)),
d=numtheory[divisors](n) minus {1, n}))
end:
a:= n-> `if`(n<2, 0, g(n$2, 1)):
MATHEMATICA
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Table[Length[Select[facs[n], OddQ@Length[#]&]], {n, 100}]
CROSSREFS
The case of set partitions (or n squarefree) is A024429. The case of partitions (or prime powers) is A027193. The remaining (even-length) factorizations are counted by A339846. A160786 counts odd-length partitions of odd numbers, ranked by A300272.
A316439 counts factorizations by product and length.
A340101 counts factorizations into odd factors.
A340102 counts odd-length factorizations into odd factors.
Expansion of e.g.f. cosh(exp(x)-1).
+10
32
1, 0, 1, 3, 8, 25, 97, 434, 2095, 10707, 58194, 338195, 2097933, 13796952, 95504749, 692462671, 5245040408, 41436754261, 340899165549, 2915100624274, 25857170687507, 237448494222575, 2253720620740362, 22078799199129799, 222987346441156585, 2319210969809731600
COMMENTS
Number of partitions of an n-element set into an even number of classes.
Let A(0) = 1, B(0) = 0; A(n+1) = Sum_{k=0..n} binomial(n,k)*B(k), B(n+1) = Sum_{k=0..n} binomial(n,k)*A(k); entry gives A sequence (cf. A024429).
REFERENCES
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 226, 5th line of table.
S. K. Ghosal, J. K. Mandal, Stirling Transform Based Color Image Authentication, Procedia Technology, 2013 Volume 10, 2013, Pages 95-104.
L. Lovasz, Combinatorial Problems and Exercises, North-Holland, 1993, pp. 15, 148.
FORMULA
a(n) = S(n, 2) + S(n, 4) + ... + S(n, 2k), where k = [ n/2 ], S(i, j) are Stirling numbers of second kind.
O.g.f.: Sum_{n>=0} x^(2*n) / Product_{k=0..2*n} (1 - k*x). - Paul D. Hanna, Sep 05 2012 G.f.: G(0)/(1+x) where G(k) = 1 - x*(2*k+1)/((2*x*k-1) - x*(2*x*k-1)/(x - (2*k+1)*(2*x*k+x-1)/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 05 2013 G.f.: G(0)/(1+2*x) where G(k) = 1 - 2*x*(k+1)/((2*x*k-1) - x*(2*x*k-1)/(x - 2*(k+1)*(2*x*k+x-1)/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 05 2013 a(n) ~ n^n / (2 * (LambertW(n))^n * exp(n+1-n/LambertW(n)) * sqrt(1+LambertW(n))). - Vaclav Kotesovec, Aug 04 2014
MAPLE
b:= proc(n, t) option remember; `if`(n=0, t, add(
b(n-j, 1-t)*binomial(n-1, j-1), j=1..n))
end:
a:= n-> b(n, 1):
with(combinat); seq((bell(n) + BellB(n, -1))/2, n = 0..20); # G. C. Greubel, Oct 09 2019
MATHEMATICA
nn=20; a=Exp[Exp[x]-1]; Range[0, nn]!CoefficientList[Series[(a+1/a)/2, {x, 0, nn}], x] (* Geoffrey Critzer, Nov 04 2012 *)
PROG
(Sage)
return add(stirling_number2(n, i) for i in range(0, n+(n+1)%2, 2))
(PARI) {a(n)=polcoeff(sum(m=0, n, x^(2*m)/prod(k=1, 2*m, 1-k*x +x*O(x^n))), n)} \\ Paul D. Hanna, Sep 05 2012 (Magma) a:= func< n | (&+[StirlingSecond(n, 2*k): k in [0..Floor(n/2)]]) >;
(GAP) List([0..25], n-> Sum([0..Int(n/2)], k-> Stirling2(n, 2*k)) ); # G. C. Greubel, Oct 09 2019
0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 4, 1, 4, 1, 4, 1, 1, 1, 10, 1, 1, 2, 4, 1, 7, 1, 8, 1, 1, 1, 13, 1, 1, 1, 10, 1, 7, 1, 4, 4, 1, 1, 24, 1, 4, 1, 4, 1, 10, 1, 10, 1, 1, 1, 22, 1, 1, 4, 16, 1, 7, 1, 4, 1, 7, 1, 38, 1, 1, 4, 4, 1
COMMENTS
a(n) is the number of permutation matrices with a negative contribution to the determinant that is the Möbius function. See A174725 for how the determinant is defined. - Mats Granvik, May 26 2017 Also the number of ordered factorizations of n into an odd number of factors > 1. The unordered case is A339890. For example, the a(n) factorizations for n = 8, 12, 24, 30, 32, 36 are: (8) (12) (24) (30) (32) (36)
(2*2*2) (2*2*3) (2*2*6) (2*3*5) (2*2*8) (2*2*9)
(2*3*2) (2*3*4) (2*5*3) (2*4*4) (2*3*6)
(3*2*2) (2*4*3) (3*2*5) (2*8*2) (2*6*3)
(2*6*2) (3*5*2) (4*2*4) (2*9*2)
(3*2*4) (5*2*3) (4*4*2) (3*2*6)
(3*4*2) (5*3*2) (8*2*2) (3*3*4)
(4*2*3) (2*2*2*2*2) (3*4*3)
(4*3*2) (3*6*2)
(6*2*2) (4*3*3)
(6*2*3)
(6*3*2)
(9*2*2)
(End)
MATHEMATICA
ordfacs[n_]:=If[n<=1, {{}}, Join@@Table[(Prepend[#1, d]&)/@ordfacs[n/d], {d, Rest[Divisors[n]]}]];
Table[Length[Select[ordfacs[n], OddQ@*Length]], {n, 100}] (* Gus Wiseman, Jan 04 2021 *)
CROSSREFS
A251683 counts ordered factorizations by product and length.
A340102 counts odd-length factorizations into odd factors.
Other cases of odd length:
- A024429 counts set partitions of odd length. - A027193 counts partitions of odd length. - A067659 counts strict partitions of odd length. - A089677 counts ordered set partitions of odd length. - A166444 counts compositions of odd length. - A332304 counts strict compositions of odd length.
Number of factorizations of n into an odd number of factors > 1, the greatest of which is odd.
+10
20
0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 2, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 1, 0, 2, 0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 0, 2, 1, 1, 1, 1, 2, 2, 0, 1, 3, 1, 0, 1, 1, 1, 2, 1, 1, 1, 0, 1, 0, 1, 1, 2, 2, 1, 1, 1, 1, 2, 0, 1, 4
EXAMPLE
The a(n) factorizations for n = 27, 84, 108, 180, 252, 360, 432:
27 2*6*7 2*6*9 4*5*9 4*7*9 5*8*9 6*8*9
3*3*3 3*4*7 3*4*9 2*2*45 6*6*7 2*4*45 2*8*27
2*2*21 2*2*27 2*6*15 2*2*63 3*8*15 4*4*27
2*2*3*3*3 3*4*15 2*6*21 4*6*15 2*2*2*6*9
2*2*3*3*5 3*4*21 2*12*15 2*2*3*4*9
2*2*3*3*7 2*2*2*5*9 2*2*2*2*27
2*3*3*4*5 2*2*2*2*3*3*3
2*2*2*3*15
MATHEMATICA
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Table[Length[Select[facs[n], OddQ[Length[#]]&&OddQ[Max@@#]&]], {n, 100}]
PROG
(PARI) A340607(n, m=n, k=0, grodd=0) = if(1==n, k, my(s=0); fordiv(n, d, if((d>1)&&(d<=m)&&(grodd||(d%2)), s += A340607(n/d, d, 1-k, bitor(1, grodd)))); (s)); \\ Antti Karttunen, Dec 13 2021
CROSSREFS
Note: Heinz numbers are given in parentheses below.
The case of odd length only is A339890. The case of all odd factors is A340102. The version for partitions is A340385. The version for prime indices is A340386. The case of odd maximum only is A340831. A316439 counts factorizations by sum and length.
A340101 counts factorizations (into odd factors = of odd numbers).
A340832 counts factorizations whose least part is odd.
Cf. A000700, A024429, A026804, A028260, A061395, A112798, A160786, A236914, A324522, A326845, A340608, A340788.
Numbers that cannot be factored into factors > 1, the least of which is odd.
+10
20
1, 2, 4, 6, 8, 10, 14, 16, 20, 22, 26, 28, 32, 34, 38, 44, 46, 52, 58, 62, 64, 68, 74, 76, 82, 86, 88, 92, 94, 104, 106, 116, 118, 122, 124, 128, 134, 136, 142, 146, 148, 152, 158, 164, 166, 172, 178, 184, 188, 194, 202, 206, 212, 214, 218, 226, 232, 236, 244
COMMENTS
Consists of 1 and all numbers that are even and have no odd divisor 1 < d <= n/d.
EXAMPLE
The sequence of terms together with their prime indices begins:
1: {} 44: {1,1,5} 106: {1,16}
2: {1} 46: {1,9} 116: {1,1,10}
4: {1,1} 52: {1,1,6} 118: {1,17}
6: {1,2} 58: {1,10} 122: {1,18}
8: {1,1,1} 62: {1,11} 124: {1,1,11}
10: {1,3} 64: {1,1,1,1,1,1} 128: {1,1,1,1,1,1,1}
14: {1,4} 68: {1,1,7} 134: {1,19}
16: {1,1,1,1} 74: {1,12} 136: {1,1,1,7}
20: {1,1,3} 76: {1,1,8} 142: {1,20}
22: {1,5} 82: {1,13} 146: {1,21}
26: {1,6} 86: {1,14} 148: {1,1,12}
28: {1,1,4} 88: {1,1,1,5} 152: {1,1,1,8}
32: {1,1,1,1,1} 92: {1,1,9} 158: {1,22}
34: {1,7} 94: {1,15} 164: {1,1,13}
38: {1,8} 104: {1,1,1,6} 166: {1,23}
For example, the factorizations of 88 are (2*2*2*11), (2*2*22), (2*4*11), (2*44), (4*22), (8*11), (88), none of which has odd minimum, so 88 is in the sequence.
MATHEMATICA
Select[Range[100], Function[n, n==1||EvenQ[n]&&Select[Rest[Divisors[n]], OddQ[#]&&#<=n/#&]=={}]]
CROSSREFS
The version looking at greatest factor is A000079. These factorization are counted by A340832. A033676 selects the maximum inferior divisor.
A055396 selects the least prime index.
- Factorizations -
A045778 counts strict factorizations.
A316439 counts factorizations by product and length.
A339890 counts factorizations of odd length.
A340653 counts balanced factorizations.
- Odd -
A000009 counts partitions into odd parts.
A024429 counts set partitions of odd length.
A026424 lists numbers with odd Omega.
A066208 lists Heinz numbers of partitions into odd parts.
A174726 counts ordered factorizations of odd length.
Cf. A026804, A027193, A050320, A244991, A340101, A340102, A340596, A340597, A340607, A340654, A340655, A340852.
Numbers that can be factored into factors > 1, the least of which is odd.
+10
19
3, 5, 7, 9, 11, 12, 13, 15, 17, 18, 19, 21, 23, 24, 25, 27, 29, 30, 31, 33, 35, 36, 37, 39, 40, 41, 42, 43, 45, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 59, 60, 61, 63, 65, 66, 67, 69, 70, 71, 72, 73, 75, 77, 78, 79, 80, 81, 83, 84, 85, 87, 89, 90, 91, 93, 95
COMMENTS
These are numbers that are odd or have an odd divisor 1 < d <= n/d.
EXAMPLE
The sequence of terms together with their prime indices begins:
3: {2} 27: {2,2,2} 48: {1,1,1,1,2}
5: {3} 29: {10} 49: {4,4}
7: {4} 30: {1,2,3} 50: {1,3,3}
9: {2,2} 31: {11} 51: {2,7}
11: {5} 33: {2,5} 53: {16}
12: {1,1,2} 35: {3,4} 54: {1,2,2,2}
13: {6} 36: {1,1,2,2} 55: {3,5}
15: {2,3} 37: {12} 56: {1,1,1,4}
17: {7} 39: {2,6} 57: {2,8}
18: {1,2,2} 40: {1,1,1,3} 59: {17}
19: {8} 41: {13} 60: {1,1,2,3}
21: {2,4} 42: {1,2,4} 61: {18}
23: {9} 43: {14} 63: {2,2,4}
24: {1,1,1,2} 45: {2,2,3} 65: {3,6}
25: {3,3} 47: {15} 66: {1,2,5}
For example, 72 is in the sequence because it has three suitable factorizations: (3*3*8), (3*4*6), (3*24).
MATHEMATICA
Select[Range[100], Function[n, n>1&&(OddQ[n]||Select[Rest[Divisors[n]], OddQ[#]&&#<=n/#&]!={})]]
CROSSREFS
The version looking at greatest factor is A057716. These factorization are counted by A340832. A033676 selects the maximum inferior divisor.
A055396 selects the least prime index.
- Factorizations -
A045778 counts strict factorizations.
A316439 counts factorizations by product and length.
A339890 counts factorizations of odd length.
A340653 counts balanced factorizations.
- Odd -
A000009 counts partitions into odd parts.
A024429 counts set partitions of odd length.
A026424 lists numbers with odd Omega.
A066208 lists Heinz numbers of partitions into odd parts.
A174726 counts ordered factorizations of odd length.
A332304 counts strict compositions of odd length.
A340692 counts partitions of odd rank.
Cf. A026804, A027193, A050320, A244991, A340101, A340102, A340596, A340597, A340607, A340654, A340655, A340852.
Number of integer partitions of n into an odd number of parts, the greatest of which is odd.
+10
18
1, 0, 2, 0, 3, 1, 6, 3, 10, 7, 18, 15, 30, 28, 51, 50, 82, 87, 134, 145, 211, 235, 331, 375, 510, 586, 779, 901, 1172, 1366, 1750, 2045, 2581, 3026, 3778, 4433, 5476, 6430, 7878, 9246, 11240, 13189, 15931, 18670, 22417, 26242, 31349, 36646, 43567, 50854
EXAMPLE
The a(3) = 2 through a(10) = 7 partitions:
3 5 321 7 332 9 532
111 311 322 521 333 541
11111 331 32111 522 721
511 531 32221
31111 711 33211
1111111 32211 52111
33111 3211111
51111
3111111
111111111
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], OddQ[Length[#]*Max[#]]&]], {n, 30}]
CROSSREFS
Partitions with odd maximum are counted by A027193, ranked by A244991. The Heinz numbers of these partitions are given by A340386. Other cases of odd length:
- A024429 counts set partitions of odd length. - A067659 counts strict partitions of odd length. - A089677 counts ordered set partitions of odd length. - A166444 counts compositions of odd length. - A174726 counts ordered factorizations of odd length. - A332304 counts strict compositions of odd length. - A339890 counts factorizations of odd length. A026804 counts partitions whose least part is odd.
A072233 counts partitions by sum and length.
A101707 counts partitions with odd rank.
A160786 counts odd-length partitions of odd numbers, ranked by A300272.
A340101 counts factorizations into odd factors.
A340102 counts odd-length factorizations into odd factors.
Let A(0) = 1, B(0) = 0; A(n+1) = Sum_{k=0..n} binomial(n,k)*B(k), B(n+1) = Sum_{k=0..n} -binomial(n,k)*A(k); entry gives A sequence (cf. A121868).
+10
15
1, 0, -1, -3, -6, -5, 33, 266, 1309, 4905, 11516, -22935, -556875, -4932512, -32889885, -174282151, -612400262, 907955295, 45283256165, 573855673458, 5397236838345, 41604258561397, 250231901787780, 756793798761989, -8425656230853383, -213091420659985440, -2990113204010882473
FORMULA
This sequence and its companion A121868 are related to the pair of constants cos(1) + sin(1) and cos(1) - sin(1) and may be viewed as generalizations of the Uppuluri-Carpenter numbers (complementary Bell numbers) A000587. Define E_2(k) = Sum_{n >= 0} (-1)^floor(n/2) * n^k/n! for k = 0,1,2,... . Then E_2(0) = cos(1) + sin(1) and E_2(1) = cos(1) - sin(1). It is easy to see that E_2(k+2) = E_2(k+1) - Sum_{i = 0..k} 2^i*binomial(k,i)*E_2(k-i) for k >= 0. Hence E_2(k) is an integral linear combination of E_2(0) and E_2(1) (a Dobinski-type relation). For example, E_2(2) = - E_2(0) + E_2(1), E_2(3) = -3*E_2(0) and E_2(4) = - 6*E_2(0) - 5*E_2(1). More examples are given below. To find the precise result, show F(k) := Sum_{n >= 0} (-1)^floor((n+1)/2)*n^k/n! satisfies the above recurrence with F(0) = E_2(1) and F(1) = -E_2(0) and then use the identity Sum_{i = 0..k} binomial(k,i)*E_2(i) = -F(k+1) to obtain E_2(k) = A121867(k) * E_2(0) - A121868(k) * E_2(1). For similar results see A143628. The decimal expansions of E_2(0) and E_2(1) are given in A143623 and A143624 respectively. (End) E.g.f.: A(x) = cos(exp(x)-1).
EXAMPLE
E_2(k) as linear combination of E_2(i), i = 0..1.
============================
..E_2(k)..|...E_2(0)..E_2(1)
============================
..E_2(2)..|....-1.......1...
..E_2(3)..|....-3.......0...
..E_2(4)..|....-6......-5...
..E_2(5)..|....-5.....-23...
..E_2(6)..|....33.....-74...
..E_2(7)..|...266....-161...
..E_2(8)..|..1309......57...
..E_2(9)..|..4905....3466...
...
(End)
MAPLE
M:=30; a:=array(0..100); b:=array(0..100); c:=array(0..100); d:=array(0..100); a[0]:=1; b[0]:=0; c[0]:=1; d[0]:=0;
for n from 1 to M do a[n]:=add(binomial(n-1, k)*b[k], k=0..n-1); b[n]:=add(binomial(n-1, k)*a[k], k=0..n-1); c[n]:=add(binomial(n-1, k)*d[k], k=0..n-1); d[n]:=-add(binomial(n-1, k)*c[k], k=0..n-1); od: ta:=[seq(a[n], n=0..M)]; tb:=[seq(b[n], n=0..M)]; tc:=[seq(c[n], n=0..M)]; td:=[seq(d[n], n=0..M)];
# Code based on Stirling transform:
stirtr:= proc(p) proc(n) option remember;
add(p(k) *Stirling2(n, k), k=0..n) end
end:
a:= stirtr(n-> (I^n + (-I)^n)/2):
PROG
(PARI) a(n) = sum(k=0, n\2, (-1)^k*stirling(n, 2*k, 2));
(Magma) [(&+[(-1)^k*StirlingSecond(n, 2*k): k in [0..Floor(n/2)]]): n in [0..30]]; // G. C. Greubel, Oct 09 2019 (Sage) [sum((-1)^k*stirling_number2(n, 2*k) for k in (0..floor(n/2))) for n in (0..30)] # G. C. Greubel, Oct 09 2019 (GAP) List([0..30], n-> Sum([0..Int(n/2)], k-> (-1)^k*Stirling2(n, 2*k)) ); # G. C. Greubel, Oct 09 2019
Search completed in 0.027 seconds
|