A279217 Expansion of Product_{k>=1} 1/(1 - x^k)^(k*(k+1)*(4*k-1)/6).

1, 1, 8, 30, 108, 357, 1205, 3838, 12083, 36896, 110828, 326281, 946086, 2700026, 7602642, 21128513, 58028309, 157588912, 423534324, 1127102360, 2971764946, 7766890826, 20131080168, 51766513279, 132117237595, 334770353022, 842462217948, 2106183375971, 5232414548275, 12920429411759, 31719180847831

Offset

0,3

Comments

Euler transform of the hexagonal pyramidal numbers (A002412).

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

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, Hexagonal Pyramidal Number

Index to sequences related to pyramidal numbers

Formula

G.f.: Product_{k>=1} 1/(1 - x^k)^(k*(k+1)*(4*k-1)/6).

a(n) ~ exp(-Zeta'(-1)/6 - Zeta(3)/(8*Pi^2) - Pi^16/(199065600000*Zeta(5)^3) - Pi^8*Zeta(3)/(6912000*Zeta(5)^2) - Zeta(3)^2/(1440*Zeta(5)) + 2*Zeta'(-3)/3 + (Pi^12/(172800000*2^(4/5)*Zeta(5)^(11/5)) + Pi^4*Zeta(3)/(7200*2^(4/5) * Zeta(5)^(6/5))) * n^(1/5) + (-Pi^8/(288000*2^(3/5)*Zeta(5)^(7/5)) - Zeta(3)/(12*2^(3/5)*Zeta(5)^(2/5))) * n^(2/5) + (Pi^4/(360*2^(2/5)*Zeta(5)^(3/5))) * n^(3/5) + 5*(Zeta(5)/2)^(1/5)/2 * n^(4/5)) * Zeta(5)^(173/1800) / (2^(26/225) * sqrt(5*Pi) * n^(1073/1800)). - Vaclav Kotesovec, Dec 08 2016

Mathematica

nmax=30; CoefficientList[Series[Product[1/(1 - x^k)^(k (k + 1)(4 k - 1)/6), {k, 1, nmax}], {x, 0, nmax}], x]

Crossrefs

Cf. A000335, A002412, A279215, A279216, A279218, A279219.

Sequence in context: A098213 A372252 A163613 * A050477 A239612 A055737

Adjacent sequences: A279214 A279215 A279216 * A279218 A279219 A279220

Keyword

nonn

Author

Ilya Gutkovskiy, Dec 08 2016

Status

approved