A321387 Expansion of Product_{k>=1} (1 + x^k)^(k^(k-1)).

1, 1, 2, 11, 74, 708, 8583, 127424, 2239965, 45514345, 1049365071, 27061132159, 771695223819, 24109698083919, 818914886275467, 30044684789498522, 1184048086192376822, 49883929845112421452, 2237287911899357657492, 106426388125032988691636, 5352033610656721914626572

Offset: 0

Author

Ilya Gutkovskiy, Nov 08 2018

Keywords

• nonn

Comments

Weigh transform of A000169.

Links

• N. J. A. Sloane, Transforms

Crossrefs

Cf. A000169, A023879, A261053, A283335, A321385, A321388.

Programs

Magma

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

Maple

a:=series(mul((1+x^k)^(k^(k-1)),k=1..100),x=0,21): seq(coeff(a,x,n),n=0..20); # Paolo P. Lava, Apr 02 2019

Mathematica

nmax = 20; CoefficientList[Series[Product[(1 + x^k)^(k^(k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d^d, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 20}]

PARI

seq(n)={Vec(exp(sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1)*d^d ) * x^k/k) + O(x*x^n)))} \\ Andrew Howroyd, Nov 09 2018

PARI

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

Formula

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

a(n) ~ n^(n-1) * (1 + exp(-1)/n + (3*exp(-1)/2 + 2*exp(-2))/n^2). - Vaclav Kotesovec, Nov 09 2018