A232976 Numerators of coefficients in expansion of Product_{k>=1} 1/(1-x^k)^(k/2).

1, 1, 11, 37, 563, 1695, 12255, 36333, 972867, 2946747, 18641221, 55674771, 691993655, 2037484683, 12296580999, 36106933117, 1708708848483, 4955653540051, 28943726818665, 83124892750711, 958302911335293, 2730521640247521, 15561772451632937, 43970981993285115, 993588138105790887, 2785544697144356207, 15601240187271712393, 43442724873393477375, 482971671644633204159

Offset

0,3

Comments

This is the square root of the g.f. for planar partitions (A000219).

Links

Table of n, a(n) for n=0..28.

Formula

a(n) / A046161(n) ~ zeta(3)^(13/72) * exp(1/24 + 3*zeta(3)^(1/3)*n^(2/3)/2) / (sqrt(6*A*Pi) * n^(49/72)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Jun 28 2025

Example

1, 1/2, 11/8, 37/16, 563/128, 1695/256, 12255/1024, 36333/2048, 972867/32768, ...

Maple

mul( 1/(1-x^k)^(k/2), k=1..29) ;
taylor(%, x=0, 29) ;
gfun[seriestolist](%) ;
map(numer, %) ; # R. J. Mathar, Dec 08 2013

Mathematica

nmax = 30; Numerator[CoefficientList[Series[Product[1/(1 - x^k)^(k/2), {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jun 28 2025 *)

Crossrefs

Denominators are A046161. Cf. A000219.

Sequence in context: A052432 A104269 A084014 * A084018 A235874 A012820

Adjacent sequences: A232973 A232974 A232975 * A232977 A232978 A232979

Keyword

nonn,frac

Author

N. J. A. Sloane, Dec 07 2013

Status

approved