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A193479 G.f. A(x) satisfies: 1+x = Sum_{n>=0} A(x)^n/sf(n), where A(x) = Sum_{n>=1} a(n)*x^n/sf(n), and sf(n) = Product_{k=0..n} k! is the superfactorial of n (A000178).

%I A193479 #9 Feb 28 2022 14:20:02
%S A193479 1,-1,5,-121,16199,-13857481,86631572159,-4470597876144961,
%T A193479 2126428452257713430399,-10305779379533133607589385601,
%U A193479 557802385738943120790269629003660799,-366846102335019802908345392106358106684889601,3169417347948517943104654704100947667168800468999705599
%N A193479 G.f. A(x) satisfies: 1+x = Sum_{n>=0} A(x)^n/sf(n), where A(x) = Sum_{n>=1} a(n)*x^n/sf(n), and sf(n) = Product_{k=0..n} k! is the superfactorial of n (A000178).
%e A193479 A(x) = x - x^2/(1!*2!) + 5*x^3/(1!*2!*3!) - 121*x^4/(1!*2!*3!*4!) + 16199*x^5/(1!*2!*3!*4!*5!) - 13857481*x^6/(1!*2!*3!*4!*5!*6!) +...+ a(n)*x^n/sf(n) +...
%e A193479 where
%e A193479 1+x = 1 + A(x) + A(x)^2/(1!*2!) + A(x)^3/(1!*2!*3!) + A(x)^4/(1!*2!*3!*4!) + A(x)^5/(1!*2!*3!*4!*5!) + A(x)^6/(1!*2!*3!*4!*5!*6!) +...+  A(x)^n/sf(n) +...
%e A193479 and sf(n) = 0!*1!*2!*3!*...*(n-1)!*n!.
%o A193479 (PARI) {a(n)=local(A=sum(m=1,n-1,a(m)*x^m/prod(k=0,m,k!))+O(x^(n+2)));
%o A193479 prod(k=0,n,k!)*polcoeff(1+x-sum(m=0,n,A^m/prod(k=0,m,k!)),n)}
%Y A193479 Cf. A000178, A193478, A193440.
%K A193479 sign
%O A193479 1,3
%A A193479 _Paul D. Hanna_, Jul 27 2011