A318493 Expansion of 1/(1 - Sum_{i>=1, j>=1} i*j*x^(i*j)).

1, 1, 5, 15, 53, 165, 561, 1807, 5993, 19586, 64491, 211466, 695101, 2281614, 7494995, 24610588, 80829373, 265437828, 871738976, 2862815763, 9401768055, 30875971366, 101399191222, 333001988025, 1093603789613, 3591473940515, 11794667169894, 38734550365835, 127207121681103, 417757532953031

Offset: 0

Author

Ilya Gutkovskiy, Aug 27 2018

Keywords

• nonn

Crossrefs

Cf. A000005, A038040, A088305, A129921, A180305, A280540, A280541.

Programs

Maple

a:=series(1/(1-add(add(i*j*x^(i*j),j=1..100),i=1..100)),x=0,30): seq(coeff(a,x,n),n=0..29); # Paolo P. Lava, Apr 02 2019

Mathematica

nmax = 29; CoefficientList[Series[1/(1 - Sum[Sum[i j x^(i j), {i, 1, nmax}], {j, 1, nmax}]), {x, 0, nmax}], x]
nmax = 29; CoefficientList[Series[1/(1 - Sum[k x^k/(1 - x^k)^2, {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 29; CoefficientList[Series[1/(1 - Sum[k DivisorSigma[0, k] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[k DivisorSigma[0, k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 29}]

Formula

G.f.: 1/(1 - Sum_{k>=1} k*x^k/(1 - x^k)^2).

G.f.: 1/(1 - Sum_{k>=1} k*d(k)*x^k), where d(k) = number of divisors of k (A000005).

a(0) = 1; a(n) = Sum_{k=1..n} A038040(k)*a(n-k).

a(n) ~ c / r^n, where r = 0.304499876501217750838861744045680232405337905509126... is the root of the equation Sum_{k>=1} k*r^k/(1 - r^k)^2 = 1 and c = 0.44152042515136849968144466258954953693306684400261343177792428746297872748... - Vaclav Kotesovec, Aug 28 2018