A090753 - OEIS

%I #19 Nov 23 2024 03:52:53

%S 1,2,2,4,16,88,600,4800,43680,443296,4949920,60217408,792134528,

%T 11200176128,169375195136,2728019576832,46626359376384,

%U 842947307334144,16073131554826752,322403473258650624,6786861273524305920

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

%C 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.

%H Vaclav Kotesovec, <a href="/A090753/b090753.txt">Table of n, a(n) for n = 0..445</a>

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

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

%F 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

%F a(n) ~ exp(-2) * n! * n. - _Vaclav Kotesovec_, Nov 23 2024

%o (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 */

%o (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)

%o for(n=0, 30, print1(a(n), ", ")) \\ _Paul D. Hanna_, Feb 04 2013

%Y Cf. A075834, A211207, A136633.

%K nonn

%O 0,2

%A _Philippe Deléham_, Feb 06 2004