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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).
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%I #9 Feb 28 2022 14:20:02

%S 1,-1,5,-121,16199,-13857481,86631572159,-4470597876144961,

%T 2126428452257713430399,-10305779379533133607589385601,

%U 557802385738943120790269629003660799,-366846102335019802908345392106358106684889601,3169417347948517943104654704100947667168800468999705599

%N 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 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 where

%e 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 and sf(n) = 0!*1!*2!*3!*...*(n-1)!*n!.

%o (PARI) {a(n)=local(A=sum(m=1,n-1,a(m)*x^m/prod(k=0,m,k!))+O(x^(n+2)));

%o prod(k=0,n,k!)*polcoeff(1+x-sum(m=0,n,A^m/prod(k=0,m,k!)),n)}

%Y Cf. A000178, A193478, A193440.

%K sign

%O 1,3

%A _Paul D. Hanna_, Jul 27 2011