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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
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
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
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Nov 07 2016
STATUS
approved