A090753 Coefficients of power series A(x) such that n-th term of A(x)^n = n!*n*x^(n-1), for n>0.

1, 2, 2, 4, 16, 88, 600, 4800, 43680, 443296, 4949920, 60217408, 792134528, 11200176128, 169375195136, 2728019576832, 46626359376384, 842947307334144, 16073131554826752, 322403473258650624, 6786861273524305920

Offset: 0

Author

Philippe Deléham, Feb 06 2004

Keywords

• nonn

Comments

At n=4 the 4th term of A(x)^4 is 4!*4x^3 = 96*x^3, as demonstrated by A(x)^4 = 1 + 8*x + 32*x^2 + 96*x^3 + 296*x^4 + ... See also A075834.

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..445

Crossrefs

Cf. A075834, A211207, A136633.

Programs

PARI

a(n)=if(n<0,0,polcoeff(x/serreverse(sum(k=1,n+1,k!*x^k,x^2*O(x^n))),n)) /* Michael Somos, Feb 14 2004 */

PARI

a(n)=local(A=1+x); for(i=1, n, A=1+2*sum(m=1, n, m^m*x^m/(A+m*x+x*O(x^n))^m)); polcoeff(A, n)
for(n=0, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Feb 04 2013

Formula

a(n) = Sum_{j=2..(n-2)} (j-1)*a(j)*a(n-j) for n>=2, with a(0)=1, a(1)=2.

Sum_{j>=0} a(j)*A090238(n-1, k+j-1) = A090238(n, k).

G.f. satisfies: A(x) = 1 + 2*Sum_{n>=1} n^n * x^n / (A(x) + n*x)^n. - Paul D. Hanna, Feb 04 2013

a(n) ~ exp(-2) * n! * n. - Vaclav Kotesovec, Nov 23 2024