A319107 Expansion of Product_{k>=1} (1 + x^k)^(sigma_1(k)-k), where sigma_1(k) = sum of divisors of k (A000203).

1, 0, 1, 1, 3, 2, 9, 5, 17, 17, 38, 33, 88, 75, 169, 181, 343, 353, 712, 728, 1348, 1518, 2591, 2898, 5025, 5615, 9259, 10866, 17160, 20111, 31775, 37264, 57130, 68782, 102663, 123698, 183793, 221708, 323077, 395325, 566079, 693248, 987086, 1210110, 1700074, 2100674, 2915549

Offset: 0

Author

Ilya Gutkovskiy, Sep 10 2018

Keywords

• nonn

Comments

Convolution of A192065 and A255528.

Weigh transform of A001065.

Links

• N. J. A. Sloane, Transforms

Crossrefs

Cf. A000203, A001065, A192065, A255528, A318784.

Programs

Maple

with(numtheory): a:=series(mul((1+x^k)^(sigma(k)-k),k=1..100),x=0,47): seq(coeff(a,x,n),n=0..46); # Paolo P. Lava, Apr 02 2019

Mathematica

nmax = 46; CoefficientList[Series[Product[(1 + x^k)^(DivisorSigma[1, k] - k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 46; CoefficientList[Series[Exp[Sum[Sum[(-1)^(k/d + 1) d (DivisorSigma[1, d] - d), {d, Divisors[k]}] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d (DivisorSigma[1, d] - d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 46}]

Formula

G.f.: Product_{k>=1} (1 + x^k)^A001065(k).

G.f.: exp(Sum_{k>=1} ( Sum_{d|k} (-1)^(k/d+1)*d*(sigma_1(d) - d) ) * x^k/k).

a(n) ~ exp(-Pi^4 / (864*(Pi^2 - 6)*Zeta(3)) - Pi^2 * n^(1/3) / (12*(2*(Pi^2 - 6)*Zeta(3))^(1/3)) + 3*((Pi^2 - 6)*Zeta(3))^(1/3) * n^(2/3) / 2^(5/3)) * ((Pi^2 - 6)*Zeta(3))^(1/6) / (2^(17/24) * sqrt(3*Pi) * n^(2/3)). - Vaclav Kotesovec, Sep 11 2018