A277212 Expansion of Product_{n>=1} (1 - x^(5*n))/(1 - x^n)^5 in powers of x.

1, 5, 20, 65, 190, 505, 1260, 2970, 6700, 14535, 30520, 62235, 123720, 240340, 457380, 854190, 1568230, 2834120, 5048140, 8871450, 15396690, 26410860, 44811440, 75254240, 125162100, 206275505, 337032360, 546183425, 878270360, 1401857550, 2221862260

Offset: 0

Author

Seiichi Manyama, Nov 07 2016

Keywords

• nonn

Comments

In general, for fixed m > 1, if g.f. = Product_{k>=1} (1 - x^(m*k))/(1 - x^k)^m, then a(n, m) ~ exp(Pi*sqrt(2*n*(m-1/m)/3)) * (m^2 - 1)^(m/4) / (2^(3*m/4 + 1/2) * 3^(m/4) * m^(m/4 + 1/2) * n^(m/4 + 1/2)). - Vaclav Kotesovec, Nov 10 2016

Examples

			G.f.: 1 + 5*x + 20*x^2 + 65*x^3 + 190*x^4 + 505*x^5 + 1260*x^6 + ...
		

Links

• Robert Israel, Table of n, a(n) for n = 0..10000
• Eric Weisstein's World of Mathematics, q-Pochhammer Symbol

Crossrefs

Cf. Expansion of Product_{n>=1} (1 - x^(k*n))/(1 - x^n)^k in powers of x: A015128 (k=2), A273845 (k=3), A274327 (k=4), this sequence (k=5), A160539 (k=7).

Cf. A109064.

Programs

Maple

N:= 100: # to get a(0)..a(N)
S:= series(mul((1-x^(5*n))/(1-x^n)^5,n=1..N),x,N+1):
seq(coeff(S,x,n),n=0..N); # Robert Israel, Nov 09 2016

Mathematica

nmax = 50; CoefficientList[Series[Product[(1 - x^(5*k))/(1 - x^k)^5, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 10 2016 *)
(QPochhammer[x^5, x^5]/QPochhammer[x, x]^5 + O[x]^40)[[3]] (* Vladimir Reshetnikov, Nov 20 2016 *)

PARI

first(n)=my(x='x); Vec(prod(k=1, n, (1-x^(5*k))/(1-x^k)^5, 1+O(x^(n+1)))) \\ Charles R Greathouse IV, Nov 07 2016

PARI

x='x+O('x^66); Vec(eta(x^5)/eta(x)^5) \\ Joerg Arndt, Nov 27 2016

Formula

G.f.: Product_{n>=1} (1 - x^(5*n))/(1 - x^n)^5.

a(n) ~ exp(4*Pi*sqrt(n/5)) / (sqrt(2) * 5^(7/4) * n^(7/4)). - Vaclav Kotesovec, Nov 10 2016

G.f.: (x^5; x^5)inf/((x; x)_inf)^5, where (a; q)_inf is the q-Pochhammer symbol. - _Vladimir Reshetnikov, Nov 20 2016

a(0) = 1, a(n) = (5/n)*Sum_{k=1..n} A285896(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 29 2017