A279219 Expansion of Product_{k>=1} 1/(1 - x^k)^(k*(k+1)*(2*k-1)/2).

1, 1, 10, 40, 155, 560, 2051, 7080, 24064, 79370, 257067, 815593, 2545201, 7812699, 23639459, 70551216, 207932549, 605611061, 1744513262, 4973116444, 14038641287, 39263308551, 108849552289, 299248060986, 816159923366, 2209102273109, 5936069692320, 15840122529455, 41987363787469, 110584436073149

Offset: 0

Author

Ilya Gutkovskiy, Dec 08 2016

Keywords

• nonn

Comments

Euler transform of the octagonal pyramidal numbers (A002414).

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
• Index to sequences related to pyramidal numbers

Crossrefs

Cf. A000335, A002414, A279215, A279216, A279217, A279218.

Programs

Mathematica

nmax=29; CoefficientList[Series[Product[1/(1 - x^k)^(k (k + 1) (2 k - 1)/2), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

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

a(n) ~ exp(-Zeta'(-1)/2 - Zeta(3)/(8*Pi^2) - Pi^16/(671846400000*Zeta(5)^3) - Pi^8*Zeta(3)/(5184000*Zeta(5)^2) - Zeta(3)^2/(240*Zeta(5)) + Zeta'(-3) + (Pi^12/(388800000*2^(3/5)*3^(1/5)*Zeta(5)^(11/5)) + Pi^4*Zeta(3)/(3600*2^(3/5) * 3^(1/5)*Zeta(5)^(6/5))) * n^(1/5) + (-Pi^8/(432000*2^(1/5)*3^(2/5)*Zeta(5)^(7/5)) - Zeta(3)/(2^(11/5)*(3*Zeta(5))^(2/5))) * n^(2/5) + (Pi^4/(180*2^(4/5)*(3*Zeta(5))^(3/5))) * n^(3/5) + ((5*(3*Zeta(5))^(1/5))/(2^(7/5))) * n^(4/5)) * (3*Zeta(5))^(9/100) / (2^(23/100) * sqrt(5*Pi) * n^(59/100)). - Vaclav Kotesovec, Dec 08 2016