A280948 Expansion of Product_{k>=1} (1 - x^(6*k)) * (1 + x^(12*k-3)) * (1 + x^(12*k-9)) / ((1 - x^(4*k-2)) * (1 - x^(2*k))).

1, 0, 2, 1, 4, 2, 7, 4, 12, 8, 20, 14, 32, 24, 50, 39, 76, 62, 114, 96, 168, 145, 244, 216, 350, 316, 496, 456, 696, 650, 968, 916, 1334, 1278, 1824, 1766, 2475, 2420, 3336, 3290, 4468, 4440, 5948, 5952, 7874, 7929, 10368, 10500, 13584, 13828, 17714

Offset

0,3

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Andrew Sills, Towards an Automation of the Circle Method, Gems in Experimental Mathematics in Contemporary Mathematics, 2010, formula S107.

Formula

a(n) ~ 2*Pi * BesselI(1, sqrt(8*n+1)*Pi/(3*sqrt(2))) / (3*sqrt(24*n+3)).

a(n) ~ exp(2*Pi*sqrt(n)/3) / (3*2^(3/2)*n^(3/4)) * (1 + (Pi/24 - 9/(16*Pi))/sqrt(n) + (Pi^2/1152 - 135/(512*Pi^2) - 15/128)/n).

Mathematica

nmax=50; CoefficientList[Series[Product[(1-x^(6*k))*(1+x^(12*k-3))*(1+x^(12*k-9))/((1-x^(4*k-2))*(1-x^(2*k))), {k, 1, nmax}], {x, 0, nmax}], x]

Crossrefs

Sequence in context: A252866 A008796 A254594 * A325345 A079966 A101707

Adjacent sequences: A280945 A280946 A280947 * A280949 A280950 A280951

Keyword

nonn

Author

Vaclav Kotesovec, Jan 11 2017

Status

approved