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

%I A179327 #26 Aug 10 2021 16:36:26
%S A179327 1,1,2,4,11,37,167,925,6164,47630,418227,4105887,44529413,528398441,
%T A179327 6807143686,94588353184,1409913624333,22437692156739,379673925360239,
%U A179327 6806484898946045,128862141334488784,2569079946351669286,53797816061915662161,1180533553597621952193
%N A179327 G.f.: Product_{n>=1} 1/(1-x^n)^((n-1)!).
%H A179327 Vaclav Kotesovec, <a href="/A179327/b179327.txt">Table of n, a(n) for n = 0..448</a>
%F A179327 Euler transform of (n-1)!.
%F A179327 G.f.: A(x) = exp( Sum_{n>=1} A062363(n)*x^n/n ) where A062363(n) = Sum_{d|n} d!.
%F A179327 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
%e A179327 G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 11*x^4 + 37*x^5 + 167*x^6 +...
%e A179327 A(x) = 1/((1-x)*(1-x^2)*(1-x^3)^2*(1-x^4)^6*(1-x^5)^24*(1-x^6)^120*...).
%e A179327 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 +...
%p A179327 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A179327       add(b(n-i*j, i-1)*binomial((i-1)!+j-1, j), j=0..n/i)))
%p A179327     end:
%p A179327 a:= n-> b(n$2):
%p A179327 seq(a(n), n=0..23);  # _Alois P. Heinz_, Aug 10 2021
%t A179327 nmax=20; CoefficientList[Series[Product[1/(1-x^k)^((k-1)!),{k,1,nmax}],{x,0,nmax}],x] (* _Vaclav Kotesovec_, Mar 14 2015 *)
%o A179327 (PARI) {a(n)=polcoeff(exp(sum(m=1,n,sumdiv(m,d,d!)*x^m/m)+x*O(x^n)),n)}
%Y A179327 Cf. A062363, A107895, A256126, A261047.
%K A179327 nonn
%O A179327 0,3
%A A179327 _Paul D. Hanna_, Jan 08 2011