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A277212 Expansion of Product_{n>=1} (1 - x^(5*n))/(1 - x^n)^5 in powers of x. 11
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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

LINKS

Robert Israel, Table of n, a(n) for n = 0..10000

Eric Weisstein's World of Mathematics, q-Pochhammer Symbol

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

EXAMPLE

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

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 *)

PROG

(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

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.

Sequence in context: A100534 A285928 A160506 * A160528 A023004 A001873

Adjacent sequences:  A277209 A277210 A277211 * A277213 A277214 A277215

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Nov 07 2016

STATUS

approved

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Last modified February 25 08:12 EST 2018. Contains 299646 sequences. (Running on oeis4.)