A112915 Recurrence: a(n) = Sum_{k=0..n-1} (k+1)*(n-k)*a(k)*a(n-k-1) for n>0, with a(0)=1.

1, 1, 4, 28, 272, 3312, 47872, 794880, 14840064, 306900736, 6953989120, 171200048128, 4548965384192, 129742326218752, 3953689388187648, 128215703582343168, 4409347536459988992, 160304460015345795072

Offset: 0

Author

Paul D. Hanna, Oct 06 2005

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..400

Crossrefs

Cf. A088716, A112916, A238223.

Programs

PARI

a(n)=if(n==0,1,sum(k=0,n-1,(k+1)*(n-k)*a(k)*a(n-k-1)))

PARI

{a(n)=local(F=1+x+x*O(x^n));for(i=1,n,F=1+x*deriv(x*F)^2); return(polcoeff(F,n,x))}

Formula

A(x) = 1 + x*G(2*x)^2, where G(x) = g.f. of A088716, such that a(n) = 2^n*A088716(n)/(n+1) for n>=0.

a(n) = 2^(n-1)*A112916(n-1) for n>0.

G.f. satisfies: A(x) = 1 + x*(d/dx[x*A(x)])^2 = 1 + x*(A(x) + x*A'(x))^2.

a(n) ~ c * n * 2^n * n!, where c = A238223 = 0.21795078944715106549... - Vaclav Kotesovec, Aug 24 2017