A280874 Expansion of Product_{k>=1} (1 - x^(6*k)) * (1 + x^k) / (1 - x^k).

1, 2, 4, 8, 14, 24, 39, 62, 96, 146, 218, 320, 463, 662, 936, 1310, 1816, 2496, 3404, 4608, 6196, 8278, 10994, 14520, 19076, 24938, 32448, 42032, 54218, 69656, 89149, 113680, 144456, 182952, 230966, 290688, 364774, 456446, 569600, 708938, 880128, 1089984

Offset

0,2

Comments

Convolution of A219601 and A000009.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000

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

Formula

a(n) ~ Pi*sqrt(2) * BesselI(1, sqrt(8*n+2)*Pi/3) / (3*sqrt(12*n+3)).

a(n) ~ exp(2*Pi*sqrt(2*n)/3) / (6*2^(3/4)*n^(3/4)) * (1 + (Pi/6 - 9/(16*Pi))/sqrt(2*n) + (Pi^2/144 - 135/(1024*Pi^2) - 15/64)/n).

Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = (1/6) * exp(Pi / 4) * 2^(11/12) * 3^(5/8) / (sqrt(2) * (1+3^(1/2)))^(1/6) = A389012. - Simon Plouffe, Sep 22 2025

Mathematica

nmax = 60; CoefficientList[Series[Product[(1-x^(6*k))*(1+x^k)/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Crossrefs

Cf. A000009, A219601.

Sequence in context: A281968 A091774 A344741 * A243815 A060046 A053801

Adjacent sequences: A280871 A280872 A280873 * A280875 A280876 A280877

Keyword

nonn

Author

Vaclav Kotesovec, Jan 09 2017

Status

approved