A294838 Expansion of Product_{k>=1} (1 + x^k)^(k*(3*k-2)).

1, 1, 8, 29, 89, 301, 915, 2763, 8040, 22910, 63776, 174174, 467448, 1233836, 3209679, 8234149, 20857621, 52206847, 129227514, 316543962, 767767628, 1844925743, 4394337797, 10379319118, 24320964976, 56557678603, 130571770387, 299357973400, 681777058604, 1542840256421, 3470045577372

Offset: 0

Author

Ilya Gutkovskiy, Nov 09 2017

Keywords

• nonn

Comments

Weigh transform of the octagonal numbers (A000567).

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

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..8270
• 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, Octagonal Number

Crossrefs

Cf. A000567, A027998, A028377, A294102, A294836, A294837.

Programs

Mathematica

nmax = 30; CoefficientList[Series[Product[(1 + x^k)^(k (3 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 (3 d - 2), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 30}]

Formula

G.f.: Product_{k>=1} (1 + x^k)^A000567(k).

a(n) ~ exp(-1800*Zeta(3)^3 / (49*Pi^8) - (9 * 2^(3/4) * 5^(5/4) * Zeta(3)^2 / (7^(5/4)*Pi^5)) * n^(1/4) - (3*sqrt(10/7) * Zeta(3) / Pi^2) * sqrt(n) + (2*(14/5)^(1/4) * Pi/3) * n^(3/4)) * 7^(1/8) / (2^(41/24) * 5^(1/8) * n^(5/8)). - Vaclav Kotesovec, Nov 10 2017

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