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 A005223 Number of Dyck paths of knight moves. (Formerly M0369) 1
 0, 0, 1, 0, 2, 2, 7, 10, 29, 52, 142, 294, 772, 1732, 4451, 10482, 26715, 64908, 165194, 409720, 1044629, 2627712, 6721492, 17079076, 43853111, 112273270, 289390434, 745262022, 1928015211, 4988699442, 12949776427, 33638741110, 87590340673 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 REFERENCES N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Harvey P. Dale, Table of n, a(n) for n = 0..1000 J. Labelle and Y.-N. Yeh, Dyck paths of knight moves, Discrete Applied Math., 24 (1989), 213-221. FORMULA G.f.=1-1/A, where A=(1+2z+sqrt(1-4z+4z^2-4z^4)-sqrt(2)*sqrt(1-4z^2-2z^4+(2z+1)sqrt(1-4z+4z^2-4z^4)))/[4z^2]. a(n) ~ (23*sqrt(2*(9-5*sqrt(3))) + sqrt(138*(7*sqrt(3)-3))) * (1+sqrt(3))^n / (184*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 10 2014 A(x) = 1 - 1/A005220(x). - Gheorghe Coserea, Jan 16 2017 MATHEMATICA CoefficientList[Series[1-1/(((1+2z+Sqrt[1-4z+4z^2-4z^4])-Sqrt[2] Sqrt[ 1-4z^2-2z^4+(2z+1)Sqrt[1-4z+4z^2-4z^4]])/(4z^2)), {z, 0, 40}], z] (* Harvey P. Dale, Oct 11 2011 *) CROSSREFS Sequence in context: A267446 A054226 A000024 * A118680 A278457 A094116 Adjacent sequences:  A005220 A005221 A005222 * A005224 A005225 A005226 KEYWORD nonn,easy,nice,walk AUTHOR EXTENSIONS More terms from Emeric Deutsch, Dec 17 2003 STATUS approved

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