A158888 G.f. satisfies: A(x) = Sum_{n>=0} x^n * A(2^n*x)^n.

1, 1, 3, 21, 305, 8785, 497089, 55504321, 12305179649, 5437293562113, 4797448178045953, 8459278545576012801, 29821007074850747998209, 210213196038821563873677313, 2963378701144932768795387346945

Offset

0,3

Comments

Compare g.f. to the g.f. C(x) of the Catalan numbers:

C(x) = Sum_{n>=0} x^n * C(x)^n.

Links

Table of n, a(n) for n=0..14.

Formula

a(n) = Sum_{k=0..n} 2^((n-k)*k) * { [x^(n-k)] A(x)^k }, where [x^(n-k)] A(x)^k denotes the coefficient of x^(n-k) in A(x)^k.

Example

G.f.: A(x) = 1 + x + 3*x^2 + 21*x^3 + 305*x^4 + 8785*x^5 +...
A(2x) = 1 + 2*x + 12*x^2 + 168*x^3 + 4880*x^4 + 281120*x^5 +...
A(4x)^2 = 1 + 8*x + 112*x^2 + 3072*x^3 + 169216*x^4 +...
A(8x)^3 = 1 + 24*x + 768*x^2 + 41984*x^3 + 4411392*x^4 +...
A(16x)^4 = 1 + 64*x + 4608*x^2 + 507904*x^3 + 102432768*x^4 +...
A(32x)^5 = 1 + 160*x + 25600*x^2 + 5734400*x^3 + 2233466880*x^4 +...

Prog

(PARI) {a(n)=local(A=1+x); for(n=2, n, A=sum(k=0, n, x^k*subst(A, x, x*2^k+x*O(x^n))^k)); polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n)=local(A=1+x); for(n=2, n, A=sum(k=0, n, x^k*subst(A, x, x*2^k+x*O(x^n))^k)); sum(k=0, n, 2^((n-k)*k)*polcoeff(A^k, n-k))}

Crossrefs

Cf. A178089, A230316.

Sequence in context: A193206 A055555 A208731 * A331583 A305532 A005329

Adjacent sequences: A158885 A158886 A158887 * A158889 A158890 A158891

Keyword

nonn

Author

Paul D. Hanna, May 01 2009

Status

approved