A318416 Expansion of Product_{i>=1, j>=1} (1 + i*j*x^(i*j)).

1, 1, 4, 10, 22, 50, 115, 231, 470, 995, 1912, 3745, 7222, 13608, 25345, 47322, 85654, 155163, 278867, 494080, 870618, 1524769, 2640527, 4549564, 7802037, 13251684, 22412317, 37706268, 63015263, 104800015, 173574936, 285694401, 468449681, 764775169, 1242535747, 2010866469, 3242127656

Offset: 0

Author

Ilya Gutkovskiy, Aug 26 2018

Keywords

• nonn

Links

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

Crossrefs

Cf. A000005, A022629, A107742, A280541, A318415.

Programs

Maple

a:=series(mul(mul(1+i*j*x^(i*j),j=1..55),i=1..55),x=0,37): seq(coeff(a,x,n),n=0..36); # Paolo P. Lava, Apr 02 2019

Mathematica

nmax = 36; CoefficientList[Series[Product[Product[(1 + i j x^(i j)), {i, 1, nmax}], {j, 1, nmax}], {x, 0, nmax}], x]
nmax = 36; CoefficientList[Series[Product[(1 + k x^k)^DivisorSigma[0, k], {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 36; CoefficientList[Series[Exp[Sum[Sum[(-d)^(k/d + 1) DivisorSigma[0, d], {d, Divisors[k]}] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-d)^(k/d + 1) DivisorSigma[0, d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 36}]
nmax = 36; s = 1 + x; Do[s *= Sum[Binomial[DivisorSigma[0, k], j]*k^j*x^(j*k), {j, 0, nmax/k}]; s = Expand[s]; s = Take[s, Min[nmax + 1, Exponent[s, x] + 1, Length[s]]];, {k, 2, nmax}]; CoefficientList[s, x] (* Vaclav Kotesovec, Aug 27 2018 *)

Formula

G.f.: Product_{k>=1} (1 + k*x^k)^tau(k), where tau = number of divisors (A000005).

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