A038629 Convolution of Catalan numbers A000108 with Catalan numbers but C(0)=1 replaced by 3.

3, 4, 9, 24, 70, 216, 693, 2288, 7722, 26520, 92378, 325584, 1158924, 4160240, 15043725, 54747360, 200360130, 736928280, 2722540590, 10098646800, 37594507860, 140415097680, 526024740930, 1976023374624, 7441754696100, 28091245875056, 106268257060308, 402815053582368

Offset

0,1

Links

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

Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.

S. Kitaev, J. Remmel and M. Tiefenbruck, Marked mesh patterns in 132-avoiding permutations I, arXiv preprint arXiv:1201.6243 [math.CO], 2012-2014. - From N. J. A. Sloane, May 09 2012

Sergey Kitaev, Jeffrey Remmel, and Mark Tiefenbruck, Quadrant Marked Mesh Patterns in 132-Avoiding Permutations II, Electronic Journal of Combinatorial Number Theory, Volume 15 #A16; also on arXiv, arXiv:1302.2274 [math.CO], 2013.

Formula

a(n) = 6*binomial(2*n, n)/(n+2) = C(n+1)+2*C(n) where C(n) are Catalan numbers.

G.f.: c(x)*(c(x)+2), where c(x) is the g.f. for Catalan numbers.

D-finite with recurrence (n+2)*a(n) -2*(n+1)*a(n-1) +4*(-2*n+3)*a(n-2)=0. - R. J. Mathar, Dec 10 2013

From Amiram Eldar, Feb 14 2023: (Start)

Sum_{n>=0} 1/a(n) = Pi/(9*sqrt(3)) + 5/9.

Sum_{n>=0} (-1)^n/a(n) = 17/75 - 22*log(phi)/(75*sqrt(5)), where phi is the golden ratio (A001622). (End)

Mathematica

Table[CatalanNumber[n + 1] + 2 CatalanNumber[n], {n, 0, 30}] (* Vincenzo Librandi, May 10 2012 *)

Prog

(Magma) [6*Binomial(2*n, n)/(n+2): n in [0..30]]; // Vincenzo Librandi, May 10 2012
(PARI) vector(100, n, n--; 6*binomial(2*n, n)/(n+2)) \\ Altug Alkan, Oct 31 2015

Crossrefs

Cf. A000108.

Sequence in context: A296265 A034921 A038222 * A354545 A247087 A270756

Adjacent sequences: A038626 A038627 A038628 * A038630 A038631 A038632

Keyword

easy,nonn

Author

Wolfdieter Lang

Status

approved