A316962 Expansion of Product_{k>=1} (1 + sigma(k)*x^k), where sigma(k) is the sum of the divisors of k (A000203).

1, 1, 3, 7, 11, 25, 51, 87, 129, 286, 462, 760, 1312, 2102, 3470, 5988, 8840, 13884, 22577, 33545, 55961, 85341, 126705, 194317, 293621, 435040, 641472, 971503, 1462483, 2108161, 3124489, 4474579, 6545809, 9561923, 13518678, 19809034, 28387625, 40286631, 57039233

Offset: 0

Author

Ilya Gutkovskiy, Jul 17 2018

Keywords

• nonn

Links

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

Crossrefs

Cf. A000203, A180305, A192065, A279786, A316961.

Programs

Maple

with(numtheory): a:=series(mul(1+sigma(k)*x^k,k=1..100),x=0,39): seq(coeff(a,x,n),n=0..38); # Paolo P. Lava, Apr 02 2019

Mathematica

nmax = 38; CoefficientList[Series[Product[(1 + DivisorSigma[1, k] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 38; CoefficientList[Series[Exp[Sum[Sum[(-1)^(k + 1) DivisorSigma[1, j]^k x^(j k)/k, {j, 1, nmax}], {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]^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 38}]

Formula

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