A294591 Expansion of Product_{k>=1} 1/((1 - x^(2*k-1))^(k*(3*k-1)/2)*(1 - x^(2*k))^(k*(3*k+1)/2)).

1, 1, 3, 8, 18, 40, 88, 184, 384, 783, 1573, 3110, 6087, 11745, 22450, 42466, 79597, 147890, 272632, 498696, 905846, 1634270, 2929804, 5220581, 9249440, 16297659, 28567571, 49825296, 86487331, 149438681, 257077485, 440378787, 751313413, 1276765557, 2161511352

Offset

0,3

Comments

Euler transform of the generalized pentagonal numbers (A001318).

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

N. J. A. Sloane, Transforms

Eric Weisstein's World of Mathematics, Pentagonal Number

Formula

G.f.: Product_{k>=1} 1/(1 - x^k)^A001318(k).

a(n) ~ exp(Pi * 2^(5/4) / (3*5^(1/4)) * n^(3/4) + 3*Zeta(3) * sqrt(5*n) / (2^(3/2) * Pi^2) + (Pi/48 - 45*Zeta(3)^2 / (8*Pi^5)) * (5*n/2)^(1/4) + 225*Zeta(3)^3 / (8*Pi^8) - 11*Zeta(3) / (64*Pi^2))/ (2^(95/48) * 5^(1/8) * n^(5/8)). - Vaclav Kotesovec, Nov 07 2017

Mathematica

nmax = 34; CoefficientList[Series[Product[1/((1 - x^(2 k - 1))^(k (3 k - 1)/2) (1 - x^(2 k))^(k (3 k + 1)/2)), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d Ceiling[d/2] Ceiling[(3 d + 1)/2]/2, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 34}]

Crossrefs

Cf. A000294, A001318, A095699, A278768, A294654, A294655, A294667, A294669.

Sequence in context: A135094 A026657 A036384 * A080692 A117080 A240135

Adjacent sequences: A294588 A294589 A294590 * A294592 A294593 A294594

Keyword

nonn

Author

Ilya Gutkovskiy, Nov 05 2017

Status

approved