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A193478 G.f. A(x) satisfies: 1/(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 A193478 #12 Feb 28 2022 14:19:34
%S A193478 1,1,5,95,9959,6270119,28519938719,1045680030158399,
%T A193478 349874346597600908159,1178635679994967168072291199,
%U A193478 44013684086180240167822552866892799,19826711369458419136710617483545735797772799,116690731684609551482643899854886684445978037938815999
%N A193478 G.f. A(x) satisfies: 1/(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 A193478 A(x) = x + x^2/(1!*2!) + 5*x^3/(1!*2!*3!) + 95*x^4/(1!*2!*3!*4!) + 9959*x^5/ (1!*2!*3!*4!*5!) + 6270119*x^6/(1!*2!*3!*4!*5!*6!) +...+ a(n)*x^n/sf(n) +...
%e A193478 where
%e A193478 1/(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 A193478 and sf(n) = 0!*1!*2!*3!*...*(n-1)!*n!.
%o A193478 (PARI) {a(n)=local(A=sum(m=1,n-1,a(m)*x^m/prod(k=0,m,k!))+O(x^(n+2)));
%o A193478 prod(k=0,n,k!)*polcoeff(1/(1-x)-sum(m=0,n,A^m/prod(k=0,m,k!)),n)}
%Y A193478 Cf. A000178, A193479, A193440.
%K A193478 nonn
%O A193478 1,3
%A A193478 _Paul D. Hanna_, Jul 27 2011