A153294 G.f.: A(x) = F(x*F(x)^2) = F(F(x)-1) where F(x) = 1 + x*F(x)^2 is the g.f. of A000108 (Catalan).

1, 1, 4, 18, 86, 427, 2180, 11373, 60380, 325259, 1773842, 9776637, 54380144, 304905223, 1721650832, 9782051362, 55888463214, 320898932595, 1850762866662, 10717217871255, 62287285235230, 363212668363520, 2124430957852380

Offset: 0

Author

Paul D. Hanna, Jan 15 2009

Keywords

• nonn

Comments

Ignoring a(0), the series reversal gives A030267 with alternating signs: 1, -4, 14, -46, 145, -444, ... - Vladimir Reshetnikov, Aug 03 2019

Examples

			G.f.: A(x) = F(x*F(x)^2) = 1 + x + 4*x^2 + 18*x^3 + 86*x^4 +... where
F(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 +...
F(x)^2 = 1 + 2*x + 5*x^2 + 14*x^3 + 42*x^4 + 132*x^5 + 429*x^6 +...
A(x)^2 = 1 + 2*x + 9*x^2 + 44*x^3 + 224*x^4 + 1170*x^5 + 6226*x^6 +...
F(x)^2*A(x)^2 = 1 + 4*x + 18*x^2 + 86*x^3 + 427*x^4 + 2180*x^5 +...
From _Peter Bala_, Jul 21 2015: (Start)
Let B(x) = (A(x) - 1)/x = Sum_{n >= 0} a(n+1)*x^n. Then 1 + x*B'(x)/B (x)  = 1 + 4*x + 20*x^2 + 106*x^3 + ... is the o.g.f. for A243585.
x*sqrt(B(x)) = x + 2*x^2 + 7*x^3 + 29*x^4 + ... is the o.g.f for A007582. (End)
		

Crossrefs

Cf. A001764, A000108; A030267, A153293, A153295, A007852, A007856, A243585.

Programs

Mathematica

a[0] = 1; a[n_] := 2^(n-1) (2n-1)!! ((Hypergeometric2F1[-1/2, -n-1, n, -4] - 1)/(n+1)! + 2 Hypergeometric2F1[1/2, -n, n+1, -4]/(n n!)); Table[a[n], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 07 2015 *)
Flatten[{1, Table[Sum[Binomial[2*k + 1, k]/(2*k + 1)*Binomial[2*(n-k) + 2*k, n-k]*2*k/(2*(n-k) + 2*k), {k, 0, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, Nov 07 2015 *)
A153294[0] := 1; A153294[n_] := (A243585[n] - A007856[n+1])/n;
Table[A153294[n], {n, 0, 22}] (* Peter Luschny, Aug 04 2019 *)

PARI

{a(n)=if(n==0,1,sum(k=0,n,binomial(2*k+1,k)/(2*k+1)*binomial(2*(n-k)+2*k,n-k)*2*k/(2*(n-k)+2*k)))}

Formula

a(n) = Sum_{k=0..n} C(2k+1,k)/(2k+1) * C(2n,n-k)*k/n for n>0 with a(0)=1.

G.f.: A(x) = [1 - sqrt(5 - 4*F(x))]/(2*F(x)-2) where F(x) = (1-sqrt(1-4x))/(2x).

G.f. satisfies: A(x) = 1 + x*F(x)^2*A(x)^2 where F(x) is the g.f. of A000108.

G.f. satisfies: A(x*G(x)) = F(x*G(x)^3) = F(G(x)-1) where G(x) = F(x*G(x)) is the g.f. of A001764 and F(x) is the g.f. of A000108.

For n > 0, a(n) = 2^(n-1)*(2*n-1)!!*((hypergeom([-1/2,-n-1], [n], -4) - 1)/(n+1)! + 2*hypergeom([1/2,-n], [n+1], -4)/(n*n!)). - Vladimir Reshetnikov, Nov 07 2015

a(n) ~ 5^(2*n + 1/2) / (sqrt(3*Pi) * n^(3/2) * 4^n). - Vaclav Kotesovec, Nov 07 2015

a(n) = (A243585(n) - A007856(n+1)) / n for n >= 1. - Peter Luschny, Aug 04 2019