A262879 Expansion of Product_{k>=1} (1+x^(3*k-2))^k.

1, 1, 0, 0, 2, 2, 0, 3, 4, 1, 4, 10, 6, 5, 16, 14, 9, 28, 32, 17, 40, 63, 41, 63, 112, 83, 94, 187, 171, 156, 301, 319, 260, 467, 580, 465, 713, 981, 818, 1095, 1627, 1452, 1682, 2584, 2510, 2632, 4047, 4266, 4162, 6181, 7054, 6685, 9396, 11423, 10753, 14132

Offset

0,5

Links

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

Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015

Formula

a(n) ~ exp(2^(-4/3) * 3^(2/3) * Zeta(3)^(1/3) * n^(2/3) + Pi^2 * n^(1/3) / (2^(2/3) * 3^(8/3) * Zeta(3)^(1/3)) - Pi^4/(2916*Zeta(3))) * Zeta(3)^(1/6) / (2^(19/36) * 3^(2/3) * sqrt(Pi) * n^(2/3)).

Mathematica

nmax=100; CoefficientList[Series[Product[(1+x^(3k-2))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax=100; CoefficientList[Series[E^Sum[(-1)^(j+1)/j*x^j/(1-x^(3j))^2, {j, 1, nmax}], {x, 0, nmax}], x]
Clear[a]; a[n_]:=a[n] = If[n==0, 1, Sum[Sum[d*{0, 2*Floor[d/6] + 1, -Floor[d/6] - 1, 0, 2*Floor[d/6] + 2, 0}[[1 + Mod[d, 6]]], {d, Divisors[j]}] * a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 100}]

Crossrefs

Cf. A026007, A027346, A035528, A262876, A262877, A262878, A262884, A262949.

Sequence in context: A077872 A300453 A239292 * A278482 A324657 A369524

Adjacent sequences: A262876 A262877 A262878 * A262880 A262881 A262882

Keyword

nonn

Author

Vaclav Kotesovec, Oct 04 2015

Status

approved