A261052 Expansion of Product_{k>=1} (1+x^k)^(k!).

1, 1, 2, 8, 31, 157, 915, 6213, 48240, 423398, 4147775, 44882107, 531564195, 6837784087, 94909482330, 1413561537884, 22482554909451, 380269771734265, 6815003300096013, 128992737080703803, 2571218642722865352, 53835084737513866662, 1181222084520177393143

Offset: 0

Author

Vaclav Kotesovec, Aug 08 2015

Keywords

• nonn

Comments

Weigh transform of the factorial numbers. - Alois P. Heinz, Jun 11 2018

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..450

Crossrefs

Cf. A000142, A107895, A168246, A261053, A026007, A027998, A248882, A102866, A256142.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(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=25; CoefficientList[Series[Product[(1+x^k)^(k!),{k,1,nmax}],{x,0,nmax}],x]

PARI

seq(n)={Vec(exp(x*Ser(dirmul(vector(n, n, n!), -vector(n, n, (-1)^n/n)))))} \\ Andrew Howroyd, Jun 22 2018

Formula

a(n) ~ n! * (1 + 1/n + 2/n^2 + 10/n^3 + 57/n^4 + 401/n^5 + 3382/n^6 + 33183/n^7 + 371600/n^8 + 4685547/n^9 + 65792453/n^10).