A268316 a(n) is the number of Dyck paths of length 4n and height n.

1, 1, 7, 57, 484, 4199, 36938, 328185, 2937932, 26457508, 239414383, 2175127695, 19827974412, 181266501290, 1661241473220, 15257624681145, 140400178555644, 1294141164447692, 11946771748196428, 110435320379615620, 1022108852175416720, 9470416604629933935

Offset: 0

Author

Gheorghe Coserea, Feb 01 2016

Keywords

• nonn
• walk

Comments

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

Examples

			For n = 2 the a(2) = 7 solutions are
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LLRLRLRR     /      \     /|\
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LRLLRRLR     /\/  \/\      |
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LLRRLLRR     /  \/  \     /  \
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LLRRLRLR     /  \/\/\     /
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                  /\      /|\
LRLRLLRR     /\/\/  \        \
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LLRLRRLR     /    \/\     /\
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LRLLRLRR     /\/    \       /\
		

Links

• Gheorghe Coserea, Table of n, a(n) for n = 0..1000.
• Gheorghe Coserea, Solutions for n=3.
• Gheorghe Coserea, Solutions for n=4.
• Gheorghe Coserea, MiniZinc model for generating solutions.

Crossrefs

Cf. A080936, A268315.

Column k=2 of A289481.

Programs

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

Mathematica

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 *)
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 *)

PARI

a(n) = binomial(4*n,n) * 2*(2*n+3)*(2*n^2+1)/((3*n+1)*(3*n+2)*(3*n+3));
vector(21, i, a(i))

Formula

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

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

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

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

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

Extensions

Added a(0)=1, adjusted b-file - N. J. A. Sloane, Dec 22 2016