%I #37 Mar 29 2023 07:44:07

%S 3,4,9,24,70,216,693,2288,7722,26520,92378,325584,1158924,4160240,

%T 15043725,54747360,200360130,736928280,2722540590,10098646800,

%U 37594507860,140415097680,526024740930,1976023374624,7441754696100,28091245875056,106268257060308,402815053582368

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

%H Vincenzo Librandi, <a href="/A038629/b038629.txt">Table of n, a(n) for n = 0..200</a>

%H Alexander Burstein and Louis W. Shapiro, <a href="https://arxiv.org/abs/2112.11595">Pseudo-involutions in the Riordan group</a>, arXiv:2112.11595 [math.CO], 2021.

%H S. Kitaev, J. Remmel and M. Tiefenbruck, <a href="http://arxiv.org/abs/1201.6243">Marked mesh patterns in 132-avoiding permutations I</a>, arXiv preprint arXiv:1201.6243 [math.CO], 2012-2014. - From _N. J. A. Sloane_, May 09 2012

%H Sergey Kitaev, Jeffrey Remmel, and Mark Tiefenbruck, <a href="http://www.emis.de/journals/INTEGERS/papers/p16/p16.Abstract.html">Quadrant Marked Mesh Patterns in 132-Avoiding Permutations II</a>, Electronic Journal of Combinatorial Number Theory, Volume 15 #A16; also on <a href="http://arxiv.org/abs/1302.2274">arXiv</a>, arXiv:1302.2274 [math.CO], 2013.

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

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

%F 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

%F From _Amiram Eldar_, Feb 14 2023: (Start)

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

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

%t Table[CatalanNumber[n + 1] + 2 CatalanNumber[n], {n, 0, 30}] (* _Vincenzo Librandi_, May 10 2012 *)

%o (Magma) [6*Binomial(2*n, n)/(n+2): n in [0..30]]; // _Vincenzo Librandi_, May 10 2012

%o (PARI) vector(100, n, n--; 6*binomial(2*n, n)/(n+2)) \\ _Altug Alkan_, Oct 31 2015

%Y Cf. A000108.

%K easy,nonn

%O 0,1

%A _Wolfdieter Lang_