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A038629
Convolution of Catalan numbers A000108 with Catalan numbers but C(0)=1 replaced by 3.
5
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
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
KEYWORD
easy,nonn
STATUS
approved