A294837 Expansion of Product_{k>=1} (1 + x^k)^(k*(5*k-3)/2).

1, 1, 7, 25, 73, 236, 688, 1994, 5573, 15272, 40896, 107526, 277999, 707209, 1774067, 4390665, 10734216, 25941541, 62022609, 146793160, 344129900, 799517074, 1841734224, 4208327222, 9542121050, 21477834062, 48005313446, 106579556936, 235107392079, 515441826521, 1123360284127, 2434346065621

Offset

0,3

Comments

Weigh transform of the heptagonal numbers (A000566).

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -n*(5*n-3)/2, g(n) = -1. - Seiichi Manyama, Nov 14 2017

Links

Seiichi Manyama, Table of n, a(n) for n = 0..8757

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

Eric Weisstein's World of Mathematics, Heptagonal Number

Formula

G.f.: Product_{k>=1} (1 + x^k)^A000566(k).

a(n) ~ 7^(1/8) * exp(2*Pi*7^(1/4) * n^(3/4) / 3^(5/4) - 9*Zeta(3) * sqrt(3*n/7) /(2*Pi^2) - 243*Zeta(3)^2 * (3*n/7)^(1/4) / (28*Pi^5) - 2187*Zeta(3)^3 / (98*Pi^8)) / (2^(15/8) * 3^(1/8) * n^(5/8)). - Vaclav Kotesovec, Nov 10 2017

a(0) = 1 and a(n) = (1/(2*n)) * Sum_{k=1..n} b(k)*a(n-k) where b(n) = Sum_{d|n} d^2*(5*d-3)*(-1)^(1+n/d). - Seiichi Manyama, Nov 14 2017

Mathematica

nmax = 31; CoefficientList[Series[Product[(1 + x^k)^(k (5 k - 3)/2), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d^2 (5 d - 3)/2, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 31}]

Crossrefs

Cf. A000566, A027998, A028377, A294102, A294836, A294838.

Sequence in context: A213569 A299269 A048477 * A117152 A155281 A155254

Adjacent sequences: A294834 A294835 A294836 * A294838 A294839 A294840

Keyword

nonn

Author

Ilya Gutkovskiy, Nov 09 2017

Status

approved