A255965 Expansion of Product_{k>=1} 1/(1-x^k)^binomial(k+6,7).

1, 1, 9, 45, 201, 819, 3357, 13329, 52215, 199686, 750733, 2774793, 10112184, 36357280, 129131448, 453379226, 1574884565, 5415956550, 18450934294, 62303210591, 208624947952, 693066815809, 2285129922950, 7480504628754, 24320897894515, 78557786077315

Offset: 0

Author

Vaclav Kotesovec, Mar 12 2015

Keywords

• nonn

Comments

In general, if g.f. = Product_{k>=1} 1/(1-x^k)^binomial(k+m-2,m-1) and m >= 1, then log(a(n)) ~ (m+1) * Zeta(m+1)^(1/(m+1)) * (n/m)^(m/(m+1)).

Links

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

Crossrefs

Cf. A000041 (m=1), A000219 (m=2), A000294 (m=3), A000335 (m=4), A000391 (m=5), A000417 (m=6), A000428 (m=7).

Programs

Mathematica

nmax=40; CoefficientList[Series[Product[1/(1-x^k)^(k*(k+1)*(k+2)*(k+3)*(k+4)*(k+5)*(k+6)/7!),{k,1,nmax}],{x,0,nmax}],x]

Formula

G.f.: exp(Sum_{k>=1} x^k/(k*(1 - x^k)^8)). - Ilya Gutkovskiy, May 28 2018