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A274327 Expansion of Product_{n>=1} (1 - x^(4*n))/(1 - x^n)^4 in powers of x. 5
1, 4, 14, 40, 104, 248, 560, 1200, 2474, 4924, 9520, 17928, 33008, 59528, 105408, 183536, 314744, 532208, 888382, 1465208, 2389808, 3857456, 6166096, 9766576, 15336816, 23888844, 36924656, 56659296, 86341664, 130710104, 196640576, 294059872, 437232746, 646561792 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Seiichi Manyama, 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^(4*n))/(1 - x^n)^4.

a(n) ~ 5*exp(Pi*sqrt(5*n/2)) / (2^(13/2) * n^(3/2)). - Vaclav Kotesovec, Nov 10 2016

G.f.: (x^4; x^4)_inf/((x; x)_inf)^4, where (a; q)_inf is the q-Pochhammer symbol. - Vladimir Reshetnikov, Nov 20 2016

a(0) = 1, a(n) = (4/n)*Sum_{k=1..n} A285895(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 29 2017

EXAMPLE

G.f.: 1 + 4*x + 14*x^2 + 40*x^3 + 104*x^4 + 248*x^5 + 560*x^6 + ...

MATHEMATICA

nmax = 50; CoefficientList[Series[Product[(1 - x^(4*k))/(1 - x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 10 2016 *)

(QPochhammer[x^4, x^4]/QPochhammer[x, x]^4 + O[x]^40)[[3]] (* Vladimir Reshetnikov, Nov 20 2016 *)

PROG

(PARI) first(n)=my(x='x); Vec(prod(k=1, n, (1-x^(4*k))/(1-x^k)^4, 1+O(x^(n+1)))) \\ Charles R Greathouse IV, Nov 07 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), this sequence (k=4), A277212 (k=5), A277283 (k=6), A160539 (k=7).

Cf. A083703.

Sequence in context: A093160 A001938 A066368 * A160463 A278680 A121593

Adjacent sequences:  A274324 A274325 A274326 * A274328 A274329 A274330

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Nov 07 2016

STATUS

approved

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Last modified April 19 11:58 EDT 2021. Contains 343114 sequences. (Running on oeis4.)