A099976 Bisection of A000984.

2, 20, 252, 3432, 48620, 705432, 10400600, 155117520, 2333606220, 35345263800, 538257874440, 8233430727600, 126410606437752, 1946939425648112, 30067266499541040, 465428353255261088, 7219428434016265740

Offset: 0

Author

N. J. A. Sloane, Nov 19 2004

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..100

Crossrefs

Cf. A000108, A000265, A000984, A001448, A002458, A063079.

Programs

Magma

[Binomial(4*n+2, 2*n+1): n in [0..20]]; // Vincenzo Librandi, May 22 2011

Maple

seq(binomial(4*n+2,2*n+1),n=0..20); # Emeric Deutsch, Dec 20 2004

Mathematica

Array[Binomial[4*# + 2, 2*# + 1] &, 20, 0] (* Paolo Xausa, Jul 11 2024 *)

Formula

a(n) = binomial(4n+2, 2n+1). - Emeric Deutsch, Dec 20 2004

G.f.: 2*sqrt(2)/sqrt(1-16*x)/sqrt(1+sqrt(1-16*x)) = 2 + 60*x/(G(0)-30*x) where G(k)= 2*x*(4*k+3)*(4*k+5) + (2*k+3)*(k+1)- 2*x*(k+1)*(2*k+3)*(4*k+7)*(4*k+9)/G(k+1); (continued fraction, Euler's 1st kind, 1-step). - Sergei N. Gladkovskii, Jul 14 2012

G.f. A(x) satisfies A(x^2) = F'(x)/F(x), where F(x) = C(x)/C(-x) and C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. - Peter Bala, May 15 2023

From R. J. Mathar, Jul 11 2024: (Start)

D-finite with recurrence n*(2*n+1)*a(n) -2*(4*n-1)*(4*n+1)*a(n-1)=0.

a(n) = 2*A002458(n).

G.f.: 2* 2F1(3/4,5/4; 3/2 ; 16*x).

Conjecture: A000265(a(n)) = A063079(n+1), odd part of a(n). (End)

a(n) / (2*n+2) = A024492(n). - R. J. Mathar, Jul 12 2024

Extensions

More terms from Emeric Deutsch, Dec 20 2004