A294841 Expansion of Product_{k>=1} (1 + x^(2*k-1))^(k*(3*k-2))*(1 + x^(2*k))^(k*(3*k+2)).

1, 1, 5, 13, 34, 87, 212, 504, 1167, 2665, 5933, 13042, 28191, 60148, 126688, 263821, 543414, 1108272, 2239182, 4484482, 8907530, 17555485, 34345465, 66724969, 128772908, 246951514, 470738283, 892159198, 1681544803, 3152656375, 5880839454, 10916463171, 20169007200, 37095527149

Offset: 0

Author

Ilya Gutkovskiy, Nov 09 2017

Keywords

• nonn

Comments

Weigh transform of the generalized octagonal numbers (A001082).

Links

• 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. A001082, A028377, A294838, A294839, A294840.

Programs

Mathematica

nmax = 33; CoefficientList[Series[Product[(1 + x^(2 k - 1))^(k (3 k - 2)) (1 + x^(2 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 (d^2 + d - Ceiling[d/2]^2), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 33}]

Formula

G.f.: Product_{k>=1} (1 + x^k)^A001082(k+1).

a(n) ~ exp(Pi/3 * (7/5)^(1/4) * 2^(3/4) * n^(3/4) + 9*Zeta(3) / (2*Pi^2) * sqrt(5*n/14) - (405*Zeta(3)^2 / (56*Pi^5) + Pi/48) * (10*n/7)^(1/4) + (6075*Zeta(3)^2 / (196*Pi^8) + 15/(224*Pi^2)) * Zeta(3)) * 7^(1/8) / (2^(9/4) * 5^(1/8) * n^(5/8)). - Vaclav Kotesovec, Nov 10 2017