A321388 Expansion of Product_{k>=1} (1 + x^k)^(k^(k-2)).

1, 1, 1, 4, 19, 144, 1443, 18295, 280918, 5069651, 105147307, 2464296222, 64402891501, 1856989724951, 58560557062508, 2004999890781363, 74069439021212783, 2936703201134924845, 124383305232306494864, 5605027085651919547476, 267759074907470856179460, 13516676464234372267564939

Offset: 0

Author

Ilya Gutkovskiy, Nov 08 2018

Keywords

• nonn

Comments

Weigh transform of A000272.

Links

• N. J. A. Sloane, Transforms

Crossrefs

Cf. A000272, A261053, A262842, A321386, A321387.

Programs

Magma

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(1 + x^k)^(k^(k-2)): k in [1..m]]) )); // G. C. Greubel, Nov 09 2018

Maple

a:=series(mul((1+x^k)^(k^(k-2)),k=1..100),x=0,22): seq(coeff(a,x,n),n=0..21); # Paolo P. Lava, Apr 02 2019

Mathematica

nmax = 21; CoefficientList[Series[Product[(1 + x^k)^(k^(k - 2)), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d^(d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 21}]

PARI

m=30; x='x+O('x^m); Vec(prod(k=1,m,(1+x^k)^(k^(k-2)))) \\ G. C. Greubel, Nov 09 2018

Formula

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

a(n) ~ n^(n-2) * (1 + exp(-1)/n + (5*exp(-1)/2 + exp(-2))/n^2). - Vaclav Kotesovec, Nov 09 2018