A014433 a(n) = Sum_{i=0..n-1} a(i)*a(n-i), with a(0) = 1 and a(1) = 4.

1, 4, 4, 20, 52, 228, 804, 3444, 13780, 59588, 253252, 1113556, 4892276, 21860260, 98055780, 444148020, 2021194260, 9257373060, 42583930500, 196811777940, 913015265460, 4251135572580, 19856669967780, 93027410579700, 436999575464532, 2057978301836868, 9713953354107844

Offset: 0

Author

Wouter Meeussen

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Programs

Maple

a := n -> `if`(n=0,1,simplify(2^(n+1)*GegenbauerC(n-1,-n,-1/4)/n)):
seq(a(n), n=0..26); # Peter Luschny, May 08 2016

Mathematica

Table[SeriesCoefficient[(1+x-Sqrt[1-2*x-15*x^2])/(2*x),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 07 2012 *)
nxt[{n_,a_,b_}]:={n+1,b,(b(2n+1)+15a(n-1))/(n+2)}; NestList[nxt,{1,1,4},30][[All,2]] (* Harvey P. Dale, Jul 07 2019 *)

PARI

x='x+O('x^66); Vec((1+x-sqrt(1-2*x-15*x^2))/(2*x)) \\ Joerg Arndt, May 04 2013

Formula

G.f.: (1+x-sqrt(1-2*x-15*x^2))/(2*x) - C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 19 2004

a(n) = (-3)^n*(hypergeom([1/2, n+1],[1],8/5)-5*hypergeom([1/2, n],[1],8/5))*(-15)^(1/2)/(10*(n+1)) for n>0. - Mark van Hoeij, Jul 02 2010

Recurrence: (n+1)*a(n) = (2*n-1)*a(n-1) + 15*(n-2)*a(n-2). - Vaclav Kotesovec, Oct 07 2012

a(n) ~ 5^(n+1/2)/(sqrt(2*Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 07 2012

a(n) = (-2)^(n+1)*C(2*n,n-1)*hypergeom([-n-1,-n+1],[-n+1/2],5/8)/n for n>=1. - Peter Luschny, May 08 2016

a(n) = 2^(n+1)*GegenbauerC(n-1,-n,-1/4)/n for n>=1. - Peter Luschny, May 08 2016

G.f.: 1 + 4*x/G(x) with G(x) = (1 - x - 4*x^2/G(x)) (continued fraction). - Nikolaos Pantelidis, Jan 09 2023