A279215 Expansion of Product_{k>=1} 1/(1 - x^k)^(k*(k+1)*(2*k+1)/6).

1, 1, 6, 20, 65, 190, 571, 1616, 4555, 12439, 33515, 88517, 230738, 592321, 1502384, 3763946, 9328899, 22880511, 55585077, 133806273, 319373068, 756124040, 1776497540, 4143489680, 9597505006, 22083821765, 50494638926, 114758996621, 259303832735, 582655202940, 1302234303910, 2895530963661, 6406348746390

Offset

0,3

Comments

Euler transform of the square pyramidal numbers (A000330).

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, Square Pyramidal Number

Index to sequences related to pyramidal numbers

Formula

G.f.: Product_{k>=1} 1/(1 - x^k)^(k*(k+1)*(2*k+1)/6).

a(n) ~ exp(Zeta'(-1)/6 - Zeta(3)/(8*Pi^2) - Pi^16/(24883200000*Zeta(5)^3) + Pi^8*Zeta(3)/(1728000*Zeta(5)^2) - Zeta(3)^2/(720*Zeta(5)) + Zeta'(-3)/3 + (Pi^12/(43200000*2^(3/5)*Zeta(5)^(11/5)) - Pi^4*Zeta(3) / (3600*2^(3/5) * Zeta(5)^(6/5))) * n^(1/5) + (-Pi^8/(144000*2^(1/5)*Zeta(5)^(7/5)) + Zeta(3)/(12*2^(1/5)*Zeta(5)^(2/5))) * n^(2/5) + Pi^4/(180*2^(4/5)*Zeta(5)^(3/5)) * n^(3/5) + 5*Zeta(5)^(1/5)/2^(7/5) * n^(4/5)) * Zeta(5)^(23/225) / (2^(29/150) * sqrt(5*Pi) * n^(271/450)). - Vaclav Kotesovec, Dec 08 2016

Mathematica

nmax=32; CoefficientList[Series[Product[1/(1 - x^k)^(k (k + 1) (2 k + 1)/6), {k, 1, nmax}], {x, 0, nmax}], x]

Crossrefs

Cf. A000330, A000335, A279216, A279217, A279218, A279219.

Sequence in context: A018808 A027107 A247307 * A318337 A264307 A050930

Adjacent sequences: A279212 A279213 A279214 * A279216 A279217 A279218

Keyword

nonn

Author

Ilya Gutkovskiy, Dec 08 2016

Status

approved