A136633 G.f.: A(x) = Series_Reversion( x / Sum_{n>=0} (n+1)!*x^n ).

1, 2, 10, 68, 544, 4832, 46312, 471536, 5055328, 56795840, 667286656, 8197599104, 105446118784, 1423627264256, 20234885027968, 303737480337152, 4827671316780544, 81385455480335360, 1455806861755411456

Offset

0,2

Links

Table of n, a(n) for n=0..18.

Formula

G.f. A(x) satisfies:

(1) A(x) = F(x*A(x)) and A(x/F(x)) = F(x);

(2) a(n) = [x^n] F(x)^n / (n+1);

where F(x) = Sum_{n>=0} (n+1)!*x^n.

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

Example

G.f.: A(x) = 1 + 2*x + 10*x^2 + 68*x^3 + 544*x^4 + 4832*x^5 + 46312*x^6 +...
Let F(x) = 1 + 2x + 6x^2 + 24x^3 + 120x^4 + 720x^5 +...+ (n+1)!*x^n +...
then A(x) = F(x*A(x)) and A(x/F(x)) = F(x);
also, a(n) = coefficient of x^n in F(x)^n divided by (n+1).
The g.f. A(x) also satisfies:
A(x) = 1 + 2*x*A(x)/(1+x*A(x)) + 2*2^2*x^2*A(x)^2/(1+2*x*A(x))^2 + 2*3^3*x^3*A(x)^3/(1+3*x*A(x))^3 + 2*4^4*x^4*A(x)^4/(1+4*x*A(x))^4 +...

Prog

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

Crossrefs

Cf. A090753, A075834, A211207.

Sequence in context: A147725 A074603 A110520 * A335729 A361769 A082580

Adjacent sequences: A136630 A136631 A136632 * A136634 A136635 A136636

Keyword

nonn

Author

Paul D. Hanna, Jan 14 2008

Status

approved