A261052 - OEIS

%I #15 Jul 02 2018 01:53:01

%S 1,1,2,8,31,157,915,6213,48240,423398,4147775,44882107,531564195,

%T 6837784087,94909482330,1413561537884,22482554909451,380269771734265,

%U 6815003300096013,128992737080703803,2571218642722865352,53835084737513866662,1181222084520177393143

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

%C Weigh transform of the factorial numbers. - _Alois P. Heinz_, Jun 11 2018

%H Alois P. Heinz, <a href="/A261052/b261052.txt">Table of n, a(n) for n = 0..450</a>

%F 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).

%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,

%p       add(binomial(i!, j)*b(n-i*j,i-1), j=0..n/i)))

%p     end:

%p a:= n-> b(n$2):

%p seq(a(n), n=0..25);  # _Alois P. Heinz_, Aug 08 2015

%t nmax=25; CoefficientList[Series[Product[(1+x^k)^(k!),{k,1,nmax}],{x,0,nmax}],x]

%o (PARI) seq(n)={Vec(exp(x*Ser(dirmul(vector(n, n, n!), -vector(n, n, (-1)^n/n)))))} \\ _Andrew Howroyd_, Jun 22 2018

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

%K nonn

%O 0,3

%A _Vaclav Kotesovec_, Aug 08 2015