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
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.
Crossrefs
Cf. A000108.
Programs
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Magma
[6*Binomial(2*n, n)/(n+2): n in [0..30]]; // Vincenzo Librandi, May 10 2012
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Mathematica
Table[CatalanNumber[n + 1] + 2 CatalanNumber[n], {n, 0, 30}] (* Vincenzo Librandi, May 10 2012 *) -
PARI
vector(100, n, n--; 6*binomial(2*n, n)/(n+2)) \\ Altug Alkan, Oct 31 2015
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)