A279218 Expansion of Product_{k>=1} 1/(1 - x^k)^(k*(k+1)*(5*k-2)/6).

1, 1, 9, 35, 131, 454, 1601, 5325, 17467, 55588, 173858, 532809, 1607056, 4769263, 13957660, 40302923, 114962909, 324157109, 904247056, 2496917319, 6829241131, 18510038697, 49741367504, 132582175873, 350655140642, 920568519505, 2399692063845, 6213105691838, 15982216140168, 40855658807127, 103814659491641

Offset

0,3

Comments

Euler transform of the heptagonal pyramidal numbers (A002413).

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, Heptagonal Pyramidal Number

Index to sequences related to pyramidal numbers

Formula

G.f.: Product_{k>=1} 1/(1 - x^k)^(k*(k+1)*(5*k-2)/6).

a(n) ~ exp(-Zeta'(-1)/3 - Zeta(3)/(8*Pi^2) - Pi^16/(388800000000*Zeta(5)^3) - Pi^8*Zeta(3)/(5400000*Zeta(5)^2) - Zeta(3)^2/(450*Zeta(5)) + 5*Zeta'(-3)/6 + (Pi^12/(270000000*2^(2/5)*5^(1/5)*Zeta(5)^(11/5)) + Pi^4*Zeta(3)/(4500*2^(2/5) * 5^(1/5)*Zeta(5)^(6/5))) * n^(1/5) + (-Pi^8/(180000*2^(4/5)*5^(2/5)*Zeta(5)^(7/5)) - Zeta(3)/(3*2^(4/5)*(5*Zeta(5))^(2/5))) * n^(2/5) + (Pi^4/(180*2^(1/5)*(5*Zeta(5))^(3/5))) * n^(3/5) + ((5*(5*Zeta(5))^(1/5))/(2^(8/5))) * n^(4/5)) * Zeta(5)^(67/720) / (2^(113/360) * 5^(293/720) * sqrt(Pi) * n^(427/720)). - Vaclav Kotesovec, Dec 08 2016

Mathematica

nmax=30; CoefficientList[Series[Product[1/(1 - x^k)^(k (k + 1) (5 k - 2)/6), {k, 1, nmax}], {x, 0, nmax}], x]

Crossrefs

Cf. A000335, A002413, A279215, A279216, A279217, A279219.

Sequence in context: A180082 A002418 A118414 * A322239 A232282 A364573

Adjacent sequences: A279215 A279216 A279217 * A279219 A279220 A279221

Keyword

nonn

Author

Ilya Gutkovskiy, Dec 08 2016

Status

approved