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Number of subwords of the form UDD in nondecreasing Dyck paths of length 2n.
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0, 0, 1, 4, 14, 45, 138, 411, 1200, 3454, 9836, 27779, 77938, 217493, 604222, 1672246, 4613030, 12689265, 34817418, 95320335, 260436588, 710278318, 1933906496, 5257545599, 14273273314, 38699274665, 104799960058, 283487736166, 766045036730, 2067997219629, 5577597593466, 15030365074659, 40470488092008
COMMENTS
A Dyck path is nondecreasing if the y-coordinates of its valleys form a nondecreasing sequence.
a(n) also represents the number of subwords of the form UUDDD in nondecreasing Dyck paths of length 2n.
FORMULA
a(n) = (2*F(2*n-3) + n*L(2*n-3) + 2*L(2*n-2) - 5*2^(n-2))/5 for n>=2, where F(n) = A000045(n) and L(n) = A000032(n). G.f.: x^2*(1-x)*(x^3-2*x^2+3*x-1)/((2*x-1)*(x^2-3*x+1)^2). - Alois P. Heinz, Nov 03 2024
MATHEMATICA
Table[If[n<2, 0, (2*Fibonacci[2n-3] + n*LucasL[2n-3]+2 LucasL[2n-2]-5*2^(n-2))/5], {n, 0, 20}]
Number of subwords of the form DDD in nondecreasing Dyck paths of length 2n.
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0, 0, 0, 1, 6, 26, 97, 333, 1085, 3411, 10448, 31376, 92773, 270907, 783003, 2243815, 6383550, 18048494, 50755897, 142067625, 396014681, 1099863867, 3044737100, 8404071596, 23135752141, 63538808311, 174120317367, 476207551183
COMMENTS
A Dyck path is nondecreasing if the y-coordinates of its valleys form a nondecreasing sequence.
FORMULA
a(n) = n*F(2*n-3) - L(2*n-2) + 2^(n-2) for n>=2, where F(n) = A000045(n) and L(n) = A000032(n). G.f.: x^3*(1 - 2*x + x^2 - x^3)/((1 - 2*x)*(1 - 3*x + x^2)^2).
MATHEMATICA
Table[If[n<2, 0, n Fibonacci[2 n-3]-LucalL[2 n-2]+2^(n-2)], {n, 0, 20}]
Powerful numbers that are divisible by the cube of a single prime.
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8, 16, 27, 32, 64, 72, 81, 108, 125, 128, 144, 200, 243, 256, 288, 324, 343, 392, 400, 500, 512, 576, 625, 675, 729, 784, 800, 968, 972, 1024, 1125, 1152, 1323, 1331, 1352, 1372, 1568, 1600, 1800, 1936, 2025, 2048, 2187, 2197, 2304, 2312, 2401, 2500, 2700, 2704, 2888, 2916
COMMENTS
Numbers whose prime factorization contains one exponent that equals 3, and all the others, if they exist, are equal to 2.
FORMULA
Sum_{n>=1} 1/a(n) = (15/Pi^2) * Sum_{p prime} 1/((p-1)*(p^2+1))) = 0.40258439321745859629... .
MATHEMATICA
q[n_] := Module[{e = Sort[FactorInteger[n][[;; , 2]]]}, e[[-1]] > 2 && (Length[e] == 1 || e[[-2]] == 2)]; With[{max = 3000}, Select[Union@ Flatten@Table[i^2 * j^3, {j, 1, max^(1/3)}, {i, 1, Sqrt[max/j^3]}], q]]
PROG
(PARI) is(k) = if(k == 1, 0, my(e = vecsort(factor(k)[, 2])); e[1] > 1 && e[#e] > 2 && (#e == 1 || e[#e - 1] == 2));
Powerful numbers that are not divisible by the cubes of more than one distinct prime.
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1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72, 81, 100, 108, 121, 125, 128, 144, 169, 196, 200, 225, 243, 256, 288, 289, 324, 343, 361, 392, 400, 441, 484, 500, 512, 529, 576, 625, 675, 676, 729, 784, 800, 841, 900, 961, 968, 972, 1024, 1089, 1125, 1152, 1156, 1225
COMMENTS
Subsequence of A377821 and first differs from it at n = 33: A377821(33) = 432 = 2^4 * 3^3 is not a term of this sequence. Numbers whose prime factorization has exponents that are all larger than 1 and no more than one exponent is larger than 2.
FORMULA
Sum_{n>=1} 1/a(n) = (15/Pi^2) * (1 + Sum_{p prime} 1/((p-1)*(p^2+1))) = 1.92240214785252516795... .
MATHEMATICA
q[n_] := Module[{e = Sort[FactorInteger[n][[;; , 2]]]}, Length[e] == 1 || e[[-2]] == 2]; With[{max = 1300}, Select[Union@ Flatten@ Table[i^2 * j^3, {j, 1, max^(1/3)}, {i, 1, Sqrt[max/j^3]}], # == 1 || q[#] &]]
PROG
(PARI) is(k) = if(k == 1, 1, my(e = vecsort(factor(k)[, 2])); e[1] > 1 && (#e == 1 || e[#e - 1] == 2));
Numbers k such that (21^k + 2^k)/23 is prime.
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COMMENTS
The definition implies that k must be a prime.
a(5) > 10^5.
LINKS
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002. Eric Weisstein's World of Mathematics, Repunit.
MATHEMATICA
Select[Prime[Range[10000]], PrimeQ[(21^# + 2^#)/23] &]
CROSSREFS
Cf. A057187, A057188, A062587, A062589, A127996, A127997, A128344, A204940, A217320, A225807, A228922, A229542, A375161, A375236, A377031.
a(n) = Sum_{k=1..n} tan(k*Pi/(1+2*n))^4.
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0, 9, 90, 371, 1044, 2365, 4654, 8295, 13736, 21489, 32130, 46299, 64700, 88101, 117334, 153295, 196944, 249305, 311466, 384579, 469860, 568589, 682110, 811831, 959224, 1125825, 1313234, 1523115, 1757196, 2017269, 2305190, 2622879, 2972320, 3355561, 3774714
REFERENCES
Shigeichi Moriguchi, Kanehisa Udagawa, Shin Hitotsumatsu, "Mathematics Formulas II", Iwanami Shoten, Publishers (1957), p. 14.
FORMULA
a(n) = n * (2*n+1) * (4*n^2+6*n-1)/3.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 4.
G.f.: x * (9 + 45*x + 11*x^2 - x^3)/(1 - x)^5.
E.g.f.: exp(x)*x*(27 + 108*x + 64*x^2 + 8*x^3)/3. - Stefano Spezia, Nov 10 2024
PROG
(PARI) a(n) = n*(2*n+1)*(4*n^2+6*n-1)/3;
(PARI) my(N=40, x='x+O('x^N)); concat(0, Vec(x*(9+45*x+11*x^2-x^3)/(1-x)^5))
Numbers that have more than one odd exponent larger than 1 in their prime factorization.
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216, 864, 1000, 1080, 1512, 1944, 2376, 2744, 2808, 3000, 3375, 3456, 3672, 4000, 4104, 4320, 4968, 5400, 6048, 6264, 6696, 6750, 7000, 7560, 7776, 7992, 8232, 8856, 9000, 9261, 9288, 9504, 9720, 10152, 10584, 10648, 10976, 11000, 11232, 11448, 11880, 12000, 12744, 13000
COMMENTS
The asymptotic density of this sequence is 1 - Product_{p prime} (1 - 1/(p^2*(p+1))) * (1 + Sum_{p prime} (1/(p^3+p^2-1))) = 0.0035024748296318122535... .
MATHEMATICA
q[n_] := Count[FactorInteger[n][[;; , 2]], _?(# > 1 && OddQ[#] &)] > 1; Select[Range[13000], q]
PROG
(PARI) is(k) = #select(x -> x>1 && x%2, factor(k)[, 2]) > 1;
Numbers that have a single odd exponent larger than 1 in their prime factorization.
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8, 24, 27, 32, 40, 54, 56, 72, 88, 96, 104, 108, 120, 125, 128, 135, 136, 152, 160, 168, 184, 189, 200, 224, 232, 243, 248, 250, 264, 270, 280, 288, 296, 297, 312, 328, 343, 344, 351, 352, 360, 375, 376, 378, 384, 392, 408, 416, 424, 432, 440, 456, 459, 472, 480, 486, 488, 500
COMMENTS
Differs from A060476 by having the terms 432, 648, 1728, ..., and not having the terms 1, 216, 256, 768, 864, ... . The asymptotic density of this sequence is Product_{p prime} (1 - 1/(p^2*(p+1))) * Sum_{p prime} (1/(p^3+p^2-1)) = 0.11498368544519741081... .
MATHEMATICA
q[n_] := Count[FactorInteger[n][[;; , 2]], _?(# > 1 && OddQ[#] &)] == 1; Select[Range[500], q]
PROG
(PARI) is(k) = #select(x -> x>1 && x%2, factor(k)[, 2]) == 1;
Expansion of e.g.f. (1/x) * Series_Reversion( x * (1 - x)^2 * exp(x) ).
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1, 1, 5, 44, 577, 10104, 222133, 5886880, 182775969, 6509571200, 261665344261, 11720054882304, 578878362625825, 31259890045425664, 1832295378792935925, 115862322601669627904, 7861907382202262095297, 569837358810005613281280, 43939338917141224534941829
FORMULA
E.g.f. A(x) satisfies A(x) = exp(-x * A(x))/(1 - x*A(x))^2.
a(n) = n! * Sum_{k=0..n} (-1)^k * (n+1)^(k-1) * binomial(3*n-k+1,n-k)/k!.
PROG
(PARI) a(n) = n!*sum(k=0, n, (-1)^k*(n+1)^(k-1)*binomial(3*n-k+1, n-k)/k!);
Expansion of e.g.f. (1/x) * Series_Reversion( x * (1 - x)^3 * exp(x) ).
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1, 2, 15, 206, 4193, 113904, 3882511, 159475280, 7672503681, 423360926720, 26362968645071, 1829066086810368, 139929538526047585, 11703312997355442176, 1062423600515479191375, 104042389901715413633024, 10933256593926589800851969, 1227201235266954603172331520
FORMULA
E.g.f. A(x) satisfies A(x) = exp(-x * A(x))/(1 - x*A(x))^3.
a(n) = n! * Sum_{k=0..n} (-1)^k * (n+1)^(k-1) * binomial(4*n-k+2,n-k)/k!.
PROG
(PARI) a(n) = n!*sum(k=0, n, (-1)^k*(n+1)^(k-1)*binomial(4*n-k+2, n-k)/k!);
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