A198916 G.f. satisfies: A(x) = Sum_{n>=0} n!*x^n*A(x)^(2*n).

1, 1, 4, 23, 158, 1212, 10058, 88811, 826982, 8085950, 82922624, 893003234, 10129641140, 121552747370, 1550460365622, 21115793548491, 308004022741254, 4817224946243142, 80703099826887368, 1444218797390453282, 27501426760092853796, 554910390616969332656

Offset

0,3

Links

Table of n, a(n) for n=0..21.

Formula

G.f.: A(x) = sqrt((1/x)*Series_Reversion( x/[Sum_{n>=0} n!*x^n]^2 )).

G.f. satisfies: A(x) = 1/(1 - x*A(x)^2/(1 - x*A(x)^2/(1 - 2*x*A(x)^2/(1 - 2*x*A(x)^2/(1 - 3*x*A(x)^2/(1 - 3*x*A(x)^2/(1 - 4*x*A(x)^2/(1 - ...)))))))), a recursive continued fraction.

G.f. satisfies: A(x/F(x)^2) = F(x) where F(x) = Sum_{n>=0} n!*x^n.

a(n) = [x^n] (Sum_{k>=0} k!*x^k)^(2*n+1) / (2*n+1).

Example

G.f.: A(x) = 1 + x + 4*x^2 + 23*x^3 + 158*x^4 + 1212*x^5 + 10058*x^6 +...
where
A(x) = 1 + x*A(x)^2 + 2!*x^2*A(x)^4 + 3!*x^3*A(x)^6 + 4!*x^4*A(x)^8 +...

Prog

(PARI) {a(n)=polcoeff((1/x*serreverse(x/sum(m=0, n, m!*x^m+x^2*O(x^n))^2))^(1/2), n)}
(PARI) /* Recursive continued fraction: */
{a(n)=local(A=1+x, CF=1+x*O(x^(n+2))); for(i=1, n, for(k=1, n+1, CF=1/(1-((n-k+1)\2+1)*x*A^2*CF)); A=CF); polcoeff(A, n)}
(PARI) {a(n)=polcoeff(sum(k=0, n, k!*x^k +x*O(x^n))^(2*n+1)/(2*n+1), n)}

Crossrefs

Cf. A088368.

Sequence in context: A007297 A263843 A326350 * A182969 A263186 A245110

Adjacent sequences: A198913 A198914 A198915 * A198917 A198918 A198919

Keyword

nonn

Author

Paul D. Hanna, Oct 31 2011

Status

approved