OFFSET
1,2
COMMENTS
Equivalently, a(n) is the number of rooted plane trees with 2n+1 nodes and height n.
LINKS
Gheorghe Coserea, Table of n, a(n) for n = 1..1000
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
EXAMPLE
For n = 2 the a(2) = 7 solutions are
/\/\/\ |
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 /\/ \ /\
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 *)
PROG
(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))
(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
CROSSREFS
KEYWORD
nonn,walk
AUTHOR
Gheorghe Coserea, Feb 01 2016
STATUS
editing