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%I #28 Apr 26 2020 06:08:36
%S 1,1,3,9,45,165,1605,6645,82245,626565,7430565,47347365,1294747365,
%T 7521768165,166051339365,2617940779365,45296384587365,400983812683365,
%U 16124008954507365,137769109363339365,6031874478387211365,98938319915632651365,1994173630452264715365,27846190369337241355365,1807180003160859492619365
%N a(1) = 1; a(n+1) = n!*Sum_{k|n} a(k)/k!.
%F E.g.f.: Sum_{n>=1} a(n) * x^n / (n!*(1 - x^n)) = Sum_{n>=1} a(n+1) * x^n / n!. - _Paul D. Hanna_, Sep 04 2014
%F E.g.f. A(x) satisfies: d/dx A(x) = 1 + A(x) + A(x^2) + A(x^3) + ... - _Ilya Gutkovskiy_, May 10 2019
%e E.g.f.: A(x) = x/(1-x) + x^2/(2!*(1-x^2)) + 3*x^3/(3!*(1-x^3)) + 9*x^4/(4!*(1-x^4)) + 45*x^5/(5!*(1-x^5)) + 165*x^6/(6!*(1-x^6)) + ... + a(n)*x^n/(n!*(1-x^n)) + ...
%e such that A(x) = x + 3*x^2/2! + 9*x^3/3! + 45*x^4/4! + 165*x^5/5! + 1605*x^6/6! + ... + a(n+1)*x^n/n! + ...
%t a[1] = 1; a[n_] := a[n] = (n-1)!*Sum[a[k]/k!, {k, Divisors[n-1]}]; Table[a[n], {n, 1, 25}] (* _Vaclav Kotesovec_, Apr 26 2020 *)
%o (PARI) {a(n)=if(n==1,1,(n-1)!*sumdiv(n-1,d,a(d)/d!))}
%o for(n=1,25,print1(a(n),", ")) \\ _Paul D. Hanna_, Sep 04 2014
%o (PARI) /* From e.g.f.: */
%o {a(n)=my(A=x);if(n==1,1,for(i=1,n, A = sum(k=1,n-1,a(k)*x^k/(k!*(1-x^k +x*O(x^n) )))); (n-1)!*polcoeff(A,n-1))}
%o for(n=1,25,print1(a(n),", ")) \\ _Paul D. Hanna_, Sep 04 2014
%o (PARI) N=33; v=vector(N); v[1]=1; for(n=1,N-1,v[n+1]=n!*sumdiv(n,k,v[k]/k!)); v \\ _Joerg Arndt_, Sep 04 2014
%K nonn
%O 1,3
%A _Leroy Quet_, Mar 22 2002
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