A294959 Expansion of Product_{k>=1} (1 + x^k)^(k*((k-2)^2+k)/2).
1, 1, 2, 8, 23, 64, 160, 397, 968, 2372, 5714, 13617, 32007, 74396, 171222, 390629, 883922, 1984631, 4423528, 9790146, 21524829, 47027558, 102135967, 220565018, 473743833, 1012274948, 2152271718, 4554344649, 9593260912, 20118418061, 42012556671, 87375161720, 181001416773
Offset: 0
Keywords
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..2000
- M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
- M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
- N. J. A. Sloane, Transforms
- Index to sequences related to polygonal numbers
Programs
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Maple
g:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(binomial(((i-2)^2+i)*i/2, j)*g(n-i*j, i-1), j=0..n/i))) end: a:= n-> g(n$2): seq(a(n), n=0..35); # Alois P. Heinz, Nov 12 2017 -
Mathematica
nmax = 32; CoefficientList[Series[Product[(1 + x^k)^(k ((k - 2)^2 + k)/2), {k, 1, nmax}], {x, 0, nmax}], x] a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d^2 ((d - 2)^2 + d)/2, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 32}]
Formula
G.f.: Product_{k>=1} (1 + x^k)^A060354(k).
a(n) ~ exp(-2401 * Pi^16 / (3499200000000 * Zeta(5)^3) + 49 * Pi^8 * Zeta(3) / (2700000 * Zeta(5)^2) - 2*Zeta(3)^2 / (25*Zeta(5)) + (-343*Pi^12 / (810000000 * 2^(3/5) * 3^(2/5) * 5^(1/5) * Zeta(5)^(11/5)) + 7*Pi^4 * Zeta(3) / (750 * 2^(3/5) * 3^(2/5) * 5^(1/5) * Zeta(5)^(6/5))) * n^(1/5) + (-49*Pi^8 / (360000 * 2^(1/5) * 3^(4/5) * 5^(2/5) * Zeta(5)^(7/5)) + (3/2)^(1/5) * Zeta(3) / (5*Zeta(5))^(2/5)) * n^(2/5) - (7*Pi^4 / (180 * 2^(4/5) * 3^(1/5) * (5*Zeta(5))^(3/5))) * n^(3/5) + (3^(2/5) * 5^(6/5) * Zeta(5)^(1/5) / 2^(12/5)) * n^(4/5)) * 3^(1/5) * Zeta(5)^(1/10) / (2^(69/80) * 5^(2/5) * sqrt(Pi) * n^(3/5)). - Vaclav Kotesovec, Nov 12 2017
Comments