%I #44 Sep 08 2022 08:46:15

%S 1,1,7,57,484,4199,36938,328185,2937932,26457508,239414383,2175127695,

%T 19827974412,181266501290,1661241473220,15257624681145,

%U 140400178555644,1294141164447692,11946771748196428,110435320379615620,1022108852175416720,9470416604629933935

%N a(n) is the number of Dyck paths of length 4n and height n.

%C Equivalently, a(n) is the number of rooted plane trees with 2n+1 nodes and height n.

%H Gheorghe Coserea, <a href="/A268316/b268316.txt">Table of n, a(n) for n = 0..1000</a>.

%H Gheorghe Coserea, <a href="/A268316/a268316_2.txt">Solutions for n=3</a>.

%H Gheorghe Coserea, <a href="/A268316/a268316_3.txt">Solutions for n=4</a>.

%H Gheorghe Coserea, <a href="/A268316/a268316.mzn.txt">MiniZinc model for generating solutions</a>.

%F a(n) = T(2n,n), where T(n,k) is defined by A080936.

%F a(n) = binomial(4*n,n) * 2*(2*n+3)*(2*n^2+1)/((3*n+1)*(3*n+2)*(3*n+3)).

%F a(n) ~ K * A268315^n / sqrt(n), where K = 8/27 * sqrt(2/(3*Pi)) = 0.13649151584...

%F G.f.: -((3F2(-3/4, -1/2, -1/4; 1/3, 2/3; 256*x/27)-1)/(4*x)) + 4/5*x*3F2(5/4, 3/2, 7/4; 7/3, 8/3; 256*x/27) + 8/3*x^2*3F2(9/4, 5/2, 11/4; 10/3, 11/3; 256*x/27). - _Benedict W. J. Irwin_, Aug 09 2016

%F Recurrence: 3*(n+1)*(2*n + 1)*(3*n + 1)*(3*n + 2)*(2*n^2 - 4*n + 3)*a(n) = 8*(2*n - 1)*(2*n + 3)*(4*n - 3)*(4*n - 1)*(2*n^2 + 1)*a(n-1). - _Vaclav Kotesovec_, Aug 10 2016

%e For n = 2 the a(2) = 7 solutions are

%e /\/\/\ |

%e LLRLRLRR / \ /|\

%e ................................

%e /\ /|\

%e LRLLRRLR /\/ \/\ |

%e ................................

%e /\ /\ /\

%e LLRRLLRR / \/ \ / \

%e ................................

%e /\ /|\

%e LLRRLRLR / \/\/\ /

%e ................................

%e /\ /|\

%e LRLRLLRR /\/\/ \ \

%e ................................

%e /\/\ /\

%e LLRLRRLR / \/\ /\

%e ................................

%e /\/\ /\

%e LRLLRLRR /\/ \ /\

%t Table[Binomial[4 n, n] 2 (2 n + 3) (2 n^2 + 1) / ((3 n + 1) (3 n + 2) (3 n + 3)), {n, 1, 25}] (* _Vincenzo Librandi_, Feb 04 2016 *)

%t Drop[CoefficientList[Series[-((-1 + HypergeometricPFQ[{-3/4, -1/2, -1/4}, {1/3, 2/3}, 256 x/27])/(4x)) + 4/5 x HypergeometricPFQ[{5/4, 3/2, 7/4}, {7/3, 8/3}, 256 x/27] + 8/3 x^2 HypergeometricPFQ[{9/4, 5/2, 11/4}, {10/3, 11/3}, 256x/27], {x, 0, 20}], x], 1] (* _Benedict W. J. Irwin_, Aug 09 2016 *)

%o (PARI)

%o a(n) = binomial(4*n,n) * 2*(2*n+3)*(2*n^2+1)/((3*n+1)*(3*n+2)*(3*n+3));

%o vector(21, i, a(i))

%o (Magma) [Binomial(4*n, n)*2*(2*n+3)*(2*n^2+1)/((3*n+1)*(3*n+2)*(3*n+3)): n in [1..30]]; // _Vincenzo Librandi_, Feb 04 2016

%Y Cf. A080936, A268315.

%Y Column k=2 of A289481.

%K nonn,walk

%O 0,3

%A _Gheorghe Coserea_, Feb 01 2016

%E Added a(0)=1, adjusted b-file - _N. J. A. Sloane_, Dec 22 2016