A338645 Expansion of Product_{k>=1} (1 + x^k)^binomial(k+2,2).

1, 3, 9, 29, 78, 207, 526, 1284, 3054, 7084, 16071, 35748, 78167, 168195, 356754, 746772, 1544145, 3157056, 6387114, 12795366, 25397760, 49977262, 97542936, 188912466, 363196750, 693424803, 1315161528, 2478648920, 4643337213, 8648452782, 16019345259, 29515269060, 54104712129

Offset: 0

Author

Ilya Gutkovskiy, May 09 2021

Keywords

• nonn

Links

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

Crossrefs

Cf. A000217, A027480, A028377, A217093, A219555, A343200, A344097, A344098.

Programs

Mathematica

nmax = 32; CoefficientList[Series[Product[(1 + x^k)^Binomial[k + 2, 2], {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[(-1)^(k/d + 1) d Binomial[d + 2, 2], {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 32}]

Formula

a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d+1) * A027480(d) ) * a(n-k).

a(n) ~ (7/15)^(1/8) * 2^(-21/8) * n^(-5/8) * exp((2/3)*(7/15)^(1/4)*Pi * n^(3/4) + 9*sqrt(15/7)*zeta(3) * sqrt(n) / (2*Pi^2) + ((5/7)^(1/4)*Pi / (2*3^(3/4)) - 1215*(15/7)^(1/4)*zeta(3)^2 / (28*Pi^5)) * n^(1/4) + 54675*zeta(3)^3 / (98*Pi^8) - 45*zeta(3) / (28*Pi^2)). - Vaclav Kotesovec, May 12 2021