A277283 Expansion of Product_{n>=1} (1 - x^(6*n))/(1 - x^n)^6 in powers of x.

1, 6, 27, 98, 315, 918, 2491, 6366, 15498, 36182, 81501, 177876, 377558, 781626, 1582173, 3137832, 6108051, 11687598, 22012816, 40855674, 74799828, 135210868, 241511115, 426570624, 745516240, 1290006276, 2211202692, 3756468658, 6327617862, 10572763842

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

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

Formula

G.f.: Product_{n>=1} (1 - x^(6*n))/(1 - x^n)^6.

G.f.: (x^6; x^6)_inf/((x; x)_inf)^6, where (a; q)_inf is the q-Pochhammer symbol. - Vladimir Reshetnikov, Nov 20 2016

a(n) ~ 35*sqrt(35) * exp(sqrt(35*n)*Pi/3) / (3456*sqrt(3)*n^2). - Vaclav Kotesovec, Nov 21 2016

Example

G.f.: 1 + 6*x + 27*x^2 + 98*x^3 + 315*x^4 + 918*x^5 + 2491*x^6 + ...

Mathematica

(QPochhammer[x^6, x^6]/QPochhammer[x, x]^6 + O[x]^40)[[3]] (* Vladimir Reshetnikov, Nov 20 2016 *)
nmax = 50; CoefficientList[Series[Product[(1 - x^(6*k))/(1 - x^k)^6, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 21 2016 *)

Prog

(PARI) first(n)=my(x='x); Vec(prod(k=1, n, (1-x^(6*k))/(1-x^k)^6, 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), A274327 (k=4), A277212 (k=5), this sequence (k=6), A160539 (k=7).

Sequence in context: A071734 A160507 A182821 * A160533 A023005 A001874

Adjacent sequences: A277280 A277281 A277282 * A277284 A277285 A277286

Keyword

nonn

Author

Seiichi Manyama, Nov 07 2016

Status

approved