A318493 - OEIS

%I #9 Apr 03 2019 02:58:32

%S 1,1,5,15,53,165,561,1807,5993,19586,64491,211466,695101,2281614,

%T 7494995,24610588,80829373,265437828,871738976,2862815763,9401768055,

%U 30875971366,101399191222,333001988025,1093603789613,3591473940515,11794667169894,38734550365835,127207121681103,417757532953031

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

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

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

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

%F 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

%p 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

%t nmax = 29; CoefficientList[Series[1/(1 - Sum[Sum[i j x^(i j), {i, 1, nmax}], {j, 1, nmax}]), {x, 0, nmax}], x]

%t nmax = 29; CoefficientList[Series[1/(1 - Sum[k x^k/(1 - x^k)^2, {k, 1, nmax}]), {x, 0, nmax}], x]

%t nmax = 29; CoefficientList[Series[1/(1 - Sum[k DivisorSigma[0, k] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]

%t a[0] = 1; a[n_] := a[n] = Sum[k DivisorSigma[0, k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 29}]

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

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Aug 27 2018