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

0,3

Comments

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

Links

Table of n, a(n) for n=0..22.

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

Example

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)

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 *)

Prog

(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)))}

Crossrefs

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

Sequence in context: A082685 A111966 A225887 * A164045 A178577 A130524

Adjacent sequences: A153291 A153292 A153293 * A153295 A153296 A153297

Keyword

nonn

Author

Paul D. Hanna, Jan 15 2009

Status

approved