Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%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