Displaying 21-30 of 39 results found.
Number of directed multigraphs with loops on 4 nodes with n arcs.
+0
5
1, 2, 11, 47, 198, 713, 2423, 7388, 21003, 55433, 137944, 324659, 729022, 1567139, 3242954, 6479759, 12547894, 23607614, 43267994, 77405064, 135435666, 232137202, 390371944, 644897542, 1047890293, 1676518363, 2643628813
FORMULA
G.f.: (x^26-x^25 + 4*x^24 + 18*x^23 + 63*x^22 + 151*x^21 + 402*x^20 + 790*x^19 + 1511*x^18 + 2353*x^17 + 3400*x^16 + 4296*x^15 + 5115*x^14 + 5266*x^13 + 5115*x^12 + 4296*x^11 + 3400*x^10 + 2353*x^9 + 1511*x^8 + 790*x^7 + 402*x^6 + 151*x^5 + 63*x^4 + 18*x^3 + 4*x^2-x + 1)/((x^4-1)^4*(x^3-1)^5*(x^2-1)^4*(x-1)^3).
MAPLE
gf:= (x^26-x^25 + 4*x^24 + 18*x^23 + 63*x^22 + 151*x^21 + 402*x^20 + 790*x^19 + 1511*x^18 + 2353*x^17 + 3400*x^16 + 4296*x^15 + 5115*x^14 + 5266*x^13 + 5115*x^12 + 4296*x^11 + 3400*x^10 + 2353*x^9 + 1511*x^8 + 790*x^7 + 402*x^6 + 151*x^5 + 63*x^4 + 18*x^3 + 4*x^2-x + 1)/((x^4-1)^4*(x^3-1)^5*(x^2-1)^4*(x-1)^3):
S:= series(gf, x, 101):
MATHEMATICA
nn = 30; n = 4; CoefficientList[Series[CycleIndex[ Join[PairGroup[SymmetricGroup[n], Ordered], Permutations[Range[n*(n - 1) + 1, n*(n - 1) + n]], 2], s] /. Table[s[i] -> 1/(1 - x^i), {i, 1, n^2 - n}], {x, 0, nn}], x] (* Geoffrey Critzer, Aug 07 2015*)
Expansion of (1+x^3)/((1-x)^2*(1-x^3)^2).
+0
4
1, 2, 3, 7, 11, 15, 24, 33, 42, 58, 74, 90, 115, 140, 165, 201, 237, 273, 322, 371, 420, 484, 548, 612, 693, 774, 855, 955, 1055, 1155, 1276, 1397, 1518, 1662, 1806, 1950, 2119, 2288, 2457, 2653, 2849, 3045, 3270, 3495, 3720, 3976, 4232, 4488, 4777, 5066, 5355, 5679
FORMULA
G.f.: (1+x^3)/((1-x)^2*(1-x^3)^2) = (1+x^3)/((1-x)^4*(1+x+x^2)^2).
a(n) = (1/2)*(-4*t^3 + (2n-7)*t^2 + (4n-1)*t +2n +2), where t = floor(n/3). - Ridouane Oudra, Oct 19 2019
MAPLE
seq(add(floor(i/3)^2, i=1..n+3), n=0..60); # Ridouane Oudra, Oct 19 2019
MATHEMATICA
a[n_] := Sum[Floor[i/3]^2, {i, 1, n+3}]; Table[a[n], {n, 0, 100}] (* Enrique Pérez Herrero, Mar 20 2012 *)
PROG
(Sage)
a, b, c, m = 0, 0, 0, 0
while True:
yield (a*(a*(2*a+9)+13)+b*(b+1)*(2*b+1)+c*(c+1)*(2*c+1)+6)//6
m = m + 1 if m < 2 else 0
if m == 0: a += 1
elif m == 1: b += 1
elif m == 2: c += 1
Table read by rows: T(n,k) is the number of strongly connected directed multigraphs with loops and no vertex of degree 0, with n arcs and k vertices, which are transitive (the existence of a path between two points implies the existence of an arc between those two points).
+0
4
1, 1, 1, 1, 1, 1, 2, 1, 6, 1, 10, 1, 19, 1, 28, 1, 1, 44, 2, 1, 60, 10, 1, 85, 31, 1, 110, 90, 1, 146, 222, 1, 182, 520, 1, 231, 1090, 1, 1, 280, 2180, 2, 1, 344, 4090, 11, 1
COMMENTS
Length of the n^th row: floor(sqrt(n)).
These graphs are reflexive (each vertex has a self-loop), so T(n,k) = sum( A139621(n-k^2,m),m=0..k)
EXAMPLE
Triangle begins:
1
1
1
1 1
1 2
1 6
1 10
1 19
1 28 1
a(n) is the number of non-unimodal sequences with n nonzero terms that arise as a convolution of sequences of binomial coefficients preceded by a finite number of ones.
+0
0
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 5, 7, 12, 16, 24, 30, 41, 50, 65, 77, 96, 112, 136, 156, 185, 210, 245, 275, 316, 352, 400, 442, 497, 546, 609, 665, 736, 800, 880, 952, 1041, 1122, 1221, 1311, 1420, 1520, 1640, 1750, 1881, 2002, 2145, 2277, 2432, 2576, 2744, 2900, 3081, 3250, 3445, 3627, 3836, 4032
COMMENTS
For integers x,y,p,q >= 0, set (s_i)_{i>=1} to be the sequence of p ones followed by the binomial coefficients C(x,j) for 0 <= j <= x followed by an infinite string of zeros, and set (t_i)_{i>=1} to be the sequence of q ones followed by the binomial coefficients C(y,j) for 0 <= j <= y followed by an infinite string of zeros. Then a(n) is the number of non-unimodal sequences (r_i)_{i>=1} where r_i = Sum_{j=1..i} s_j*t_{i-j} for some(s_i) and (t_i) such that x + y + p + q + 1 = n.
Let T be a rooted tree created by identifying the root vertices of two broom graphs. a(n) is the number of trees T on n vertices whose poset of connected, vertex-induced subgraphs is not rank unimodal.
FORMULA
a(n+10) = 2*(Sum_{i=1..n/2} floor(i*(i+4)/4)) - floor(n^2/16) for n even.
a(n+10) = 2*(Sum_{i=1..(n-1)/2} floor(i(i+4)/4)) - floor((n-1)^2/16) + floor((n+1)*(n+9)/16) for n odd.
MATHEMATICA
Table[If[EvenQ[n], 2*(Sum[Floor[i(i+4)/4], {i, 0, (n/2)}]) - Floor[n^2/16], 2*(Sum[Floor[i(i+4)/4], {i, 0, (n-1)/2}]) - Floor[(n-1)^2/16] + Floor[(n+1)(n+9)/16]], {n, 0, 40}]
Number of 3-element subsets of S = {1..n} whose sum is odd.
+0
1
0, 0, 0, 0, 2, 4, 10, 16, 28, 40, 60, 80, 110, 140, 182, 224, 280, 336, 408, 480, 570, 660, 770, 880, 1012, 1144, 1300, 1456, 1638, 1820, 2030, 2240, 2480, 2720, 2992, 3264, 3570, 3876, 4218, 4560, 4940, 5320, 5740, 6160, 6622, 7084, 7590, 8096, 8648, 9200
COMMENTS
There are two cases: n is odd and n is even.
Let n be an odd integer and n > 3, the sum of 3 integers is odd when all of them are odd or one is odd and the others are even. Number of ways to choose 3 odd numbers: C((n+1)/2, 3). Number of ways to choose 2 even numbers and 1 odd: C((n-1)/2, 2)*C((n+1)/2, 1). Total number of ways: C((n+1)/2, 3) + C((n-1)/2, 2)*C((n+1)/2,1).
Let n be an even integer and n > 3. Number of ways to choose 3 odd numbers: C(n/2, 3). Number of ways to choose 2 even numbers and 1 odd: C(n/2, 2)*C(n/2, 1). Total number of ways: C(n/2, 3) + C(n/2, 2)*C(n/2, 1).
Take a chessboard of n X n unit squares in which the a1 square is black. a(n) is the number of composite squares having white unit squares on their vertices. For the number of composite squares having black unit squares on their vertices see A005993. - Ivan N. Ianakiev, Aug 19 2018
FORMULA
a(n) = C((n+1)/2, 3) + C((n-1)/2, 2)*C((n+1)/2,1) when n is odd.
a(n) = C(n/2, 3) + C(n/2, 2)*C(n/2, 1) when n is even.
a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6) for n>5.
a(n) = n*(n - 1)*(n - 2)/12 for n even.
a(n) = (n - 1)*(n + 1)*(n - 3)/12 for n odd.
G.f.: 2*x^4 / ((1-x)^4*(1+x)^2). (End)
a(n) = ((-1)^n)*(-1+n)*(3 - 3*(-1)^n - 4*((-1)^n)*n + 2*((-1)^n)*n^2)/24. - Ivan N. Ianakiev, Aug 19 2018
EXAMPLE
For n = 5 then a(5) = 4. The subsets are: {1, 2, 4}, {1, 3, 5}, {2, 3, 4}, {2, 4, 5}.
MATHEMATICA
Table[Binomial[(n + #)/2, 3] + Binomial[(n - #)/2, 2] Binomial[(n + #)/2, 1] &@ Boole@ OddQ@ n, {n, 0, 49}] (* or *)
CoefficientList[Series[2 x^4/((1 - x)^4*(1 + x)^2), {x, 0, 49}], x] (* Michael De Vlieger, Jan 07 2017 *)
PROG
(PARI) concat(vector(4), Vec(2*x^4 / ((1-x)^4*(1+x)^2) + O(x^60))) \\ Colin Barker, Dec 28 2016
Number T(n,k) of k-element subsets of [n] having an even sum; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
+0
20
1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 2, 4, 6, 3, 0, 1, 3, 6, 10, 9, 3, 0, 1, 3, 9, 19, 19, 9, 3, 1, 1, 4, 12, 28, 38, 28, 12, 4, 1, 1, 4, 16, 44, 66, 60, 40, 20, 5, 0, 1, 5, 20, 60, 110, 126, 100, 60, 25, 5, 0, 1, 5, 25, 85, 170, 226, 226, 170, 85, 25, 5, 1, 1, 6, 30, 110, 255, 396, 452, 396, 255, 110, 30, 6, 1
COMMENTS
Row n is symmetric if and only if n mod 4 in {0,3} (or if T(n,n) = 1).
FORMULA
T(n,k) = Sum_{j=0..floor((n+1)/4)} C(ceiling(n/2),2*j) * C(floor(n/2),k-2*j).
Sum_{k=0..n} k * T(n,k) = A057711(n-1) for n>0. Sum_{k=0..n} (k+1) * T(n,k) = A087447(n) + [n=2].
EXAMPLE
T(5,0) = 1: {}.
T(5,1) = 2: {2}, {4}.
T(5,2) = 4: {1,3}, {1,5}, {2,4}, {3,5}.
T(5,3) = 6: {1,2,3}, {1,2,5}, {1,3,4}, {1,4,5}, {2,3,5}, {3,4,5}.
T(5,4) = 3: {1,2,3,4}, {1,2,4,5}, {2,3,4,5}.
T(5,5) = 0.
T(7,7) = 1: {1,2,3,4,5,6,7}.
Triangle T(n,k) begins:
1;
1, 0;
1, 1, 0;
1, 1, 1, 1;
1, 2, 2, 2, 1;
1, 2, 4, 6, 3, 0;
1, 3, 6, 10, 9, 3, 0;
1, 3, 9, 19, 19, 9, 3, 1;
1, 4, 12, 28, 38, 28, 12, 4, 1;
1, 4, 16, 44, 66, 60, 40, 20, 5, 0;
1, 5, 20, 60, 110, 126, 100, 60, 25, 5, 0;
1, 5, 25, 85, 170, 226, 226, 170, 85, 25, 5, 1;
1, 6, 30, 110, 255, 396, 452, 396, 255, 110, 30, 6, 1;
MAPLE
b:= proc(n, s) option remember; expand(
`if`(n=0, s, b(n-1, s)+x*b(n-1, irem(s+n, 2))))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 1)):
seq(T(n), n=0..16);
MATHEMATICA
Flatten[Table[Sum[Binomial[Ceiling[n/2], 2j]Binomial[Floor[n/2], k-2j], {j, 0, Floor[(n+1)/4]}], {n, 0, 10}, {k, 0, n}]] (* Indranil Ghosh, Feb 26 2017 *)
PROG
(PARI) a(n, k)=sum(j=0, floor((n+1)/4), binomial(ceil(n/2), 2*j)*binomial(floor(n/2), k-2*j));
tabl(nn)={for(n=0, nn, for(k=0, n, print1(a(n, k), ", "); ); print(); ); } \\ Indranil Ghosh, Feb 26 2017
CROSSREFS
Columns k=0..10 give (offsets may differ): A000012, A004526, A002620, A005993, A005994, A032092, A032093, A018211, A018212, A282077, A282078. Lower diagonals T(n+j,n) for j=1..10 give: A004525(n+1), A282079, A228705, A282080, A282081, A282082, A282083, A282084, A282085, A282086.
The number of trees with 4 nodes labeled by positive integers, where each tree's label sum is n.
+0
2
2, 4, 10, 17, 30, 44, 67, 91, 126, 163, 213, 265, 333, 403, 491, 582, 693, 807, 944, 1084, 1249, 1418, 1614, 1814, 2044, 2278, 2544, 2815, 3120, 3430, 3777, 4129, 4520, 4917, 5355, 5799, 6287, 6781, 7321, 7868, 8463, 9065, 9718, 10378, 11091, 11812, 12588, 13372, 14214, 15064
COMMENTS
Computed by the sum over the A000055(4)=2 shapes of the trees: the linear graph of the n-Butane, and the star graph of (1)-Methyl-Propane.
FORMULA
G.f.: x^4*(2+2*x+2*x^2+x^3+x^4)/((1+x)^2*(x-1)^4*(1+x+x^2) ).
EXAMPLE
a(4)=2 because there is a linear tree with all labels equal 1 and the star tree with all labels equal to 1.
MAPLE
x^4*(2+2*x+2*x^2+x^3+x^4)/(1+x)^2/(x-1)^4/(1+x+x^2) ;
taylor(%, x=0, 80) ;
gfun[seriestolist](%) ;
Array read by rows: T(n,k) is the number of directed multigraphs with loops with n arcs, k vertices, and no vertex of degree 0.
+0
7
1, 1, 1, 5, 4, 1, 1, 9, 21, 16, 4, 1, 1, 18, 71, 108, 71, 22, 4, 1, 1, 27, 194, 491, 557, 326, 101, 22, 4, 1, 1, 43, 476, 1903, 3353, 3062, 1587, 497, 111, 22, 4, 1, 1, 59, 1030, 6298, 16644, 22352, 17035, 7982, 2433, 555, 111, 22, 4, 1, 1, 84, 2095, 18823, 72064
COMMENTS
Length of the n^th row: 2n.
FORMULA
T(n,1) = 1 if n > 0.
T(n,2n) = 1 if n > 0.
T(n,2n-1) = 4 if n >= 2.
T(n,2n-k) = A144047(k) for n large enough (conjecturally, n >= 2k is enough). T(n,2) = (n^3 + 6*n^2 + 11*n - 6)/12 + ((n+2)/4)[n even]. (the bracket means that the second term is added if and only if n is even). - Benoit Jubin, Mar 31 2012
EXAMPLE
1, 1;
1, 5, 4, 1;
1, 9, 21, 16, 4, 1;
1, 18, 71, 108, 71, 22, 4, 1;
1, 27, 194, 491, 557, 326, 101, 22, 4, 1;
1, 43, 476, 1903, 3353, 3062, 1587, 497, 111, 22, 4, 1;
1, 59, 1030, 6298, 16644, 22352, 17035, 7982, 2433, 555, 111, 22, 4, 1;
Number of planar partitions of n with trace 4.
+0
1
1, 2, 6, 14, 33, 64, 127, 228, 404, 672, 1100, 1724, 2661, 3974, 5849, 8402, 11911, 16556, 22751, 30772, 41198, 54436, 71283, 92316, 118609, 150950, 190753, 239090, 297783, 368236, 452782, 553240, 672532, 812980, 978211, 1171144, 1396235
COMMENTS
Also number of partitions of n objects of 2 colors into 4 parts, each part containing at least one black object.
REFERENCES
G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976 (Ch. XI, exercise 5 and Ch. XII, exercise 5).
FORMULA
G.f.: q^4*(q^12+q^10+2*q^9+4*q^8+2*q^7+4*q^6+2*q^5+4*q^4+2*q^3+q^2+1) / ((-1+q^4)^2*(-1+q^3)^2*(-1+q^2)^2*(-1+q)^2).
Triangle T(n,k) composed of the squares min(n,k)^2.
+0
7
1, 1, 1, 1, 4, 1, 1, 4, 4, 1, 1, 4, 9, 4, 1, 1, 4, 9, 9, 4, 1, 1, 4, 9, 16, 9, 4, 1, 1, 4, 9, 16, 16, 9, 4, 1, 1, 4, 9, 16, 25, 16, 9, 4, 1, 1, 4, 9, 16, 25, 25, 16, 9, 4, 1
EXAMPLE
Replacing each term in A003983 by its square, we get: {1},
{1, 1},
{1, 4, 1},
{1, 4, 4, 1},
{1, 4, 9, 4, 1},
{1, 4, 9, 9, 4, 1},
{1, 4, 9, 16, 9, 4, 1},
{1, 4, 9, 16, 16, 9, 4, 1},
{1, 4, 9, 16, 25, 16, 9, 4, 1},
{1, 4, 9, 16, 25, 25, 16, 9, 4, 1},
{1, 4, 9, 16, 25, 36, 25, 16, 9, 4, 1}
MATHEMATICA
Clear[p, n, i];
p[x_, n_] = Sum[x^i*If[i ==Floor[n/2] && Mod[n, 2] == 0, 0, If[i <= Floor[n/2], 2*i + 1, -(2*(n - i) + 1)]], {i, 0, n}]/(1 - x);
Table[CoefficientList[FullSimplify[p[x, n]], x], {n, 1, 11}];
Flatten[%]
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