A227824 G.f. satisfies: A(x) = Sum_{n>=0} x^n*A(x)^n * Sum_{k=0..n} binomial(n,k)^2 * x^k*(1-x)^(n-k).

1, 1, 2, 7, 24, 86, 330, 1311, 5326, 22070, 92940, 396466, 1709610, 7440200, 32636590, 144146831, 640500188, 2861175670, 12841853052, 57883546774, 261905659756, 1189161029092, 5416356944248, 24741552146026, 113317361529586, 520265301736892, 2394041095608960, 11039387236631796

Offset

0,3

Comments

Compare to the trivial identity for the Catalan function C(x) = 1 + x*C(x)^2:

C(x) = Sum_{n>=0} x^n*C(x)^n * Sum_{k=0..n} binomial(n,k)*x^k*(1-x)^(n-k).

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..170

Formula

a(n) ~ c * d^n / n^(3/2), where d = 4.871479127250632..., c = 0.4392903421166... . - Vaclav Kotesovec, Jul 05 2014

Example

G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 24*x^4 + 86*x^5 + 330*x^6 + 1311*x^7 +...
where g.f. A(x) satisfies:
A(x) = 1 + x*A(x)*((1-x) + x)
+ x^2*A(x)^2*((1-x)^2 + 2^2*x*(1-x) + x^2)
+ x^3*A(x)^3*((1-x)^3 + 3^2*x*(1-x)^2 + 3^2*x^2*(1-x) + x^3)
+ x^4*A(x)^4*((1-x)^4 + 4^2*x*(1-x)^3 + 6^2*x^2*(1-x)^2 + 4^2*x^3*(1-x) + x^4)
+ x^5*A(x)^5*((1-x)^5 + 5^2*x*(1-x)^4 + 10^2*x^2*(1-x)^3 + 10^2*x^3*(1-x)^2 + 5^2*x^4*(1-x) + x^5) +...

Prog

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^m*A^m*sum(k=0, m, binomial(m, k)^2*x^k*(1-x)^(m-k)) +x*O(x^n)));; polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Sequence in context: A256938 A150389 A183876 * A270490 A104625 A221454

Adjacent sequences: A227821 A227822 A227823 * A227825 A227826 A227827

Keyword

nonn

Author

Paul D. Hanna, Jul 31 2013

Status

approved