A261053 Expansion of Product_{k>=1} (1+x^k)^(k^k).

1, 1, 4, 31, 289, 3495, 51268, 891152, 17926913, 409907600, 10499834497, 297793199060, 9262502810645, 313457634240463, 11464902463397642, 450646709610954343, 18943070964019019671, 847932498252050293971, 40266255926484893366914, 2021845081107882645459639

Offset: 0

Author

Vaclav Kotesovec, Aug 08 2015

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..385

Crossrefs

Cf. A023880, A261052, A026007, A027998, A248882, A102866, A256142.

Programs

Magma

m:=20; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(1+x^k)^(k^k): k in [1..(m+2)]]))); // G. C. Greubel, Nov 08 2018

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(i^i, j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 08 2015

Mathematica

nmax=20; CoefficientList[Series[Product[(1+x^k)^(k^k),{k,1,nmax}],{x,0,nmax}],x]

PARI

m=20; x='x+O('x^m); Vec(prod(k=1,m, (1+x^k)^(k^k))) \\ G. C. Greubel, Nov 08 2018

Formula

a(n) ~ n^n * (1 + exp(-1)/n + (exp(-1)/2 + 4*exp(-2))/n^2).

G.f.: exp(Sum_{k>=1} ( Sum_{d|k} (-1)^(k/d+1)*d^(d+1) ) * x^k/k). - Ilya Gutkovskiy, Nov 08 2018