A269796 a(n) = 4^n * A000108(n+1).
1, 8, 80, 896, 10752, 135168, 1757184, 23429120, 318636032, 4402970624, 61641588736, 872465563648, 12463793766400, 179478630236160, 2602440138424320, 37965009078190080, 556820133146787840, 8205770383215820800, 121445401671594147840, 1804331681977970196480, 26900945076762464747520
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..500
- Samuele Giraudo, Pluriassociative algebras II: The polydendriform operad and related operads, arXiv:1603.01394 [math.CO], 2016. Cf. 2.1.10.
Programs
-
Magma
[4^n*Catalan(n+1): n in [0..25]]; // Vincenzo Librandi, Apr 25 2016
-
Mathematica
Table[4^n CatalanNumber[n + 1], {n, 0, 20}] (* Bruno Berselli, Mar 07 2016 *) -
PARI
cat(n) = binomial(2*n,n)/(n+1); a(n) = 4^n*cat(n+1);
-
Sage
[4^n*catalan_number(n+1) for n in (0..20)] # Bruno Berselli, Mar 07 2016
Formula
G.f.: (1 - sqrt(1 - 16*x) - 8*x)/(32*x^2). - Bruno Berselli, Mar 07 2016
a(n) = -2^(4*n + 3)*binomial(n + 1/2, -3/2), after Peter Luschny in A000108. - Bruno Berselli, Mar 07 2016
From Amiram Eldar, May 15 2022: (Start)
Sum_{n>=0} 1/a(n) = 52/75 + 512*arcsin(1/4)/(75*sqrt(15)).
Sum_{n>=0} (-1)^n/a(n) = 164/289 + 1536*arcsinh(1/4)/(289*sqrt(17)). (End)
From Karol A. Penson, Aug 26 2025: (Start)
O.g.f.: exp(Sum_{n>=1} A098430(n)*x^n/n).
O.g.f.: exp(Sum_{n>=1} 4^n*(2*n)!*x^n/(n*(n!)^2)).
a(n) = 4^n*binomial(2*n + 2, n + 1)/(n + 2).
a(n) = Integral_{x=0..16} (x^n*sqrt(x)*sqrt(1 - x/16)/(8*Pi)) dx. (End)