A193440 -id:A193440 - cp's OEIS frontend

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).

1, 1, 5, 95, 9959, 6270119, 28519938719, 1045680030158399, 349874346597600908159, 1178635679994967168072291199, 44013684086180240167822552866892799, 19826711369458419136710617483545735797772799, 116690731684609551482643899854886684445978037938815999

Offset: 1

Paul D. Hanna, Jul 27 2011

nonn

			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) +...
where
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) +...
and sf(n) = 0!*1!*2!*3!*...*(n-1)!*n!.

Cf. A000178, A193479, A193440.

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

Original entry on oeis.org

1, -1, 5, -121, 16199, -13857481, 86631572159, -4470597876144961, 2126428452257713430399, -10305779379533133607589385601, 557802385738943120790269629003660799, -366846102335019802908345392106358106684889601, 3169417347948517943104654704100947667168800468999705599

Offset: 1

Paul D. Hanna, Jul 27 2011

sign

			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) +...
where
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) +...
and sf(n) = 0!*1!*2!*3!*...*(n-1)!*n!.

Cf. A000178, A193478, A193440.

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

cp's OEIS Frontend

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).

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