A279216 Expansion of Product_{k>=1} 1/(1 - x^k)^(k^2*(k+1)/2).

1, 1, 7, 25, 86, 269, 862, 2606, 7812, 22704, 64989, 182356, 504414, 1373694, 3693367, 9804435, 25733084, 66808578, 171719539, 437183839, 1103143657, 2760037810, 6850400668, 16873338215, 41260373472, 100196920196, 241712863504, 579416535973, 1380517695672, 3270075208145, 7702580246941

Offset: 0

Author

Ilya Gutkovskiy, Dec 08 2016

Keywords

• nonn

Comments

Euler transform of the pentagonal pyramidal numbers (A002411).

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, Pentagonal Pyramidal Number
• Index to sequences related to pyramidal numbers

Crossrefs

Cf. A000335, A002411, A279215, A279217, A279218, A279219.

Programs

Mathematica

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

Formula

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

a(n) ~ exp(-Zeta(3)/(8*Pi^2) - Pi^16/(83980800000*Zeta(5)^3) + Zeta'(-3)/2 + (Pi^12/(97200000*2^(2/5)*3^(1/5)*Zeta(5)^(11/5))) * n^(1/5) + (-Pi^8/(108000*2^(4/5)*3^(2/5)*Zeta(5)^(7/5))) * n^(2/5) + (Pi^4/(180*2^(1/5)*(3*Zeta(5))^(3/5))) * n^(3/5) + ((5*(3*Zeta(5))^(1/5))/(2^(8/5))) * n^(4/5)) * (3*Zeta(5))^(119/1200) / (2^(181/600) * sqrt(5*Pi) * n^(719/1200)). - Vaclav Kotesovec, Dec 08 2016