A227824 - OEIS

%I #9 Jul 05 2014 07:21:13

%S 1,1,2,7,24,86,330,1311,5326,22070,92940,396466,1709610,7440200,

%T 32636590,144146831,640500188,2861175670,12841853052,57883546774,

%U 261905659756,1189161029092,5416356944248,24741552146026,113317361529586,520265301736892,2394041095608960,11039387236631796

%N 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).

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

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

%H Vaclav Kotesovec, <a href="/A227824/b227824.txt">Table of n, a(n) for n = 0..170</a>

%F a(n) ~ c * d^n / n^(3/2), where d = 4.871479127250632..., c = 0.4392903421166... . - _Vaclav Kotesovec_, Jul 05 2014

%e G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 24*x^4 + 86*x^5 + 330*x^6 + 1311*x^7 +...

%e where g.f. A(x) satisfies:

%e A(x) = 1 + x*A(x)*((1-x) + x)

%e + x^2*A(x)^2*((1-x)^2 + 2^2*x*(1-x) + x^2)

%e + x^3*A(x)^3*((1-x)^3 + 3^2*x*(1-x)^2 + 3^2*x^2*(1-x) + x^3)

%e + 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)

%e + 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) +...

%o (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)}

%o for(n=0,30,print1(a(n),", "))

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jul 31 2013