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%I #16 Dec 02 2021 19:48:29
%S 1,1,2,8,62,862,19492,656224,30739676,1906807004,151002453464,
%T 14846381034784,1772922018732328,252631570039665832,
%U 42329528274029082608,8237406877267427867648,1842215469973381977889808,469160036709398319115207696,134976328490030629922214893344
%N Expansion of e.g.f.: A(x) = exp( Sum_{n>=1} (n-1)!*x^n/n ).
%H G. C. Greubel, <a href="/A159476/b159476.txt">Table of n, a(n) for n = 0..250</a>
%F a(n) = (n-1)!*Sum_{k=1..n} (k-1)!*a(n-k)/(n-k)! for n > 0 with a(0)=1.
%F a(n) ~ (n-1)!^2. - _Vaclav Kotesovec_, Jul 10 2018
%e E.g.f.: A(x) = 1 + x + 2*x^2/2! + 8*x^3/3! + 62*x^4/4! + 862*x^5/5! + ...
%e log(A(x)) = x + x^2/2 + 2!*x^3/3 + 3!*x^4/4 + 4!*x^5/5 + 5!*x^6/6 + ...
%p a:= proc(n) option remember; `if`(n=0, 1, add(
%p a(n-i)*binomial(n-1, i-1)*(i-1)!^2, i=1..n))
%p end:
%p seq(a(n), n=0..20); # _Alois P. Heinz_, Aug 13 2019
%t a:= CoefficientList[Series[Exp[Sum[(n - 1)!*x^n/n, {n, 1, 500}]], {x, 0, 35}], x]; Table[a[[n]]*(n - 1)!, {n, 1, 30}] (* _G. C. Greubel_, Jul 09 2018 *)
%o (PARI) {a(n)=n!*polcoeff(exp(sum(k=1,n,(k-1)!*x^k/k)+x*O(x^n)),n)}
%o (PARI) {a(n)=if(n==0,1,(n-1)!*sum(k=1,n,(k-1)!*a(n-k)/(n-k)!))}
%Y Cf. A158876, A293847.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Apr 15 2009
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