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A158876 Expansion of e.g.f.: exp( Sum_{n>=1} (n-1)! * x^n ).

%I A158876 #20 Sep 08 2022 08:45:43
%S A158876 1,1,3,19,217,4041,113611,4532683,244208049,17085010897,1504881245971,
%T A158876 162835665686211,21219897528855433,3276502399914104089,
%U A158876 591351260856215820507,123322423833602768272891,29423834155886520870184801,7963056392690313008566254753
%N A158876 Expansion of e.g.f.: exp( Sum_{n>=1} (n-1)! * x^n ).
%H A158876 Alois P. Heinz, <a href="/A158876/b158876.txt">Table of n, a(n) for n = 0..253</a>
%F A158876 a(n) = (n-1)! * Sum_{k=1..n} k! * a(n-k) / (n-k)! for n>0 with a(0)=1.
%F A158876 E.g.f. A(x) satisfies:
%F A158876 (1) A'(x)/A(x) = Sum_{k>=0} (n+1)! * x^n.
%F A158876 (2) A(x) = exp(x + x^2 * A'(x)/A(x)).
%F A158876 Let F(x) = Sum_{n>=0} n! * x^n, then
%F A158876 (3) [x^n] A(x)^n * (2 - F(x)) = 0 for n > 0,
%F A158876 (4) [x^n] A(x) * (n + 1 - F(x)) = 0 for n > 0. - _Paul D. Hanna_, May 26 2018
%F A158876 a(n) ~ n! * (n-1)!. - _Vaclav Kotesovec_, Aug 01 2017
%e A158876 E.g.f.: A(x) = 1 + x + 3*x^2/2! + 19*x^3/3! + 217*x^4/4! +...
%e A158876 log(A(x)) = x + x^2 + 2!*x^3 + 3!*x^4 +...+ (n-1)!*x^n +....
%p A158876 m:=20; S:=series( exp(add((j-1)!*x^j, j=1..m+2)), x, m+1): seq(j!*coeff(S, x, j), j=0..m); # _G. C. Greubel_, Mar 04 2020
%t A158876 With[{m = 20}, CoefficientList[Series[Exp[Sum[(j-1)!*x^j, {j, m+2}]], {x, 0, m}], x]*Range[0, m]!] (* _G. C. Greubel_, Mar 04 2020 *)
%o A158876 (PARI) {a(n)=if(n==0,1,(n-1)!*sum(k=1,n,k!*a(n-k)/(n-k)!))}
%o A158876 for(n=0,20,print1(a(n),", "))
%o A158876 (PARI) {a(n)=n!*polcoeff(exp(sum(k=1,n,(k-1)!*x^k)+x*O(x^n)),n)}
%o A158876 for(n=0,20,print1(a(n),", "))
%o A158876 (Magma) m:=20; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!(Laplace( Exp( &+[Factorial(j-1)*x^j: j in [1..m+2]] ) ))); // _G. C. Greubel_, Mar 04 2020
%o A158876 (Sage)
%o A158876 m=20
%o A158876 def A158876_list(prec):
%o A158876     P.<x> = PowerSeriesRing(QQ, prec)
%o A158876     return P( exp(sum(factorial(j-1)*x^j for j in (1..m+2))) ).list()
%o A158876 a=A158876_list(m+1); [factorial(n)*a[n] for n in (0..m)] # _G. C. Greubel_, Mar 04 2020
%Y A158876 Cf. A000142, A122949.
%K A158876 nonn
%O A158876 0,3
%A A158876 _Paul D. Hanna_, Apr 13 2009