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 A005222 Number of Dyck paths of knight moves. (Formerly M3234) 1
 1, 0, 1, 0, 4, 4, 18, 26, 86, 158, 462, 976, 2665, 6082, 16040, 38338, 99536, 244880, 631923, 1583796, 4081939, 10358670, 26728731, 68425494, 176964795, 455967376, 1182454137, 3061954102, 7962768190, 20702327552, 53983118006, 140817757006 (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 Vaclav Kotesovec, Table of n, a(n) for n = 0..1000 Vaclav Kotesovec, Recurrence (of order 11) J. Labelle and Y.-N. Yeh, Dyck paths of knight moves, Discrete Applied Math., 24 (1989), 213-221. FORMULA G.f.: A+z^4A^3/(1-zA)^2, 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) ~ c * (1+sqrt(3))^n / n^(3/2), where c = sqrt(341*sqrt(3) - 225 + 3*sqrt(46*(197*sqrt(3) - 22))) / (4*sqrt(23*Pi)) = 0.794168381329... - Vaclav Kotesovec, Feb 29 2016 A(x) = x^2*A005220(x)*A005221(x) + x*A005221(x)^2 + A005220(x). - Gheorghe Coserea, Jan 16 2017 MATHEMATICA A[x_] = (s*(r-1+x-x^3) + x*(1+x)*(3+r*(x-1) + x*(6*x-5)))/(4*x^3) /. s -> Sqrt[2]*Sqrt[1+r-2*x*(2*x+x^3-r)] /. r -> Sqrt[1-4*x*(1-x+x^3)]; A[x] + O[x]^32 // CoefficientList[#, x]& (* Jean-François Alcover, Mar 26 2017, after Gheorghe Coserea *) CROSSREFS Sequence in context: A086448 A128090 A119948 * A214166 A214187 A214238 Adjacent sequences:  A005219 A005220 A005221 * A005223 A005224 A005225 KEYWORD nonn,easy,nice,walk AUTHOR EXTENSIONS More terms from Emeric Deutsch, Dec 17 2003 STATUS approved

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Last modified February 22 23:03 EST 2019. Contains 320411 sequences. (Running on oeis4.)