A179327 G.f.: Product_{n>=1} 1/(1-x^n)^((n-1)!).

1, 1, 2, 4, 11, 37, 167, 925, 6164, 47630, 418227, 4105887, 44529413, 528398441, 6807143686, 94588353184, 1409913624333, 22437692156739, 379673925360239, 6806484898946045, 128862141334488784, 2569079946351669286, 53797816061915662161, 1180533553597621952193

Offset: 0

Author

Paul D. Hanna, Jan 08 2011

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 11*x^4 + 37*x^5 + 167*x^6 +...
A(x) = 1/((1-x)*(1-x^2)*(1-x^3)^2*(1-x^4)^6*(1-x^5)^24*(1-x^6)^120*...).
log(A(x)) = x + 3*x^2/2 + 7*x^3/3 + 27*x^4/4 + 121*x^5/5 + 729*x^6/6 + 5041*x^7/7 + 40347*x^8/8 +...+ A062363(n)*x^n/n +...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..448

Crossrefs

Cf. A062363, A107895, A256126, A261047.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1)*binomial((i-1)!+j-1, j), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..23);  # Alois P. Heinz, Aug 10 2021

Mathematica

nmax=20; CoefficientList[Series[Product[1/(1-x^k)^((k-1)!),{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Mar 14 2015 *)

PARI

{a(n)=polcoeff(exp(sum(m=1,n,sumdiv(m,d,d!)*x^m/m)+x*O(x^n)),n)}

Formula

Euler transform of (n-1)!.

G.f.: A(x) = exp( Sum_{n>=1} A062363(n)*x^n/n ) where A062363(n) = Sum_{d|n} d!.

a(n) ~ (n-1)! * (1 + 1/n + 3/n^2 + 11/n^3 + 50/n^4 + 278/n^5 + 1860/n^6 + 14793/n^7 + 138166/n^8 + 1494034/n^9 + 18422609/n^10), for coefficients see A256126. - Vaclav Kotesovec, Mar 14 2015