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 A005221 Number of Dyck paths of knight moves. (Formerly M2371) 2
 0, 0, 1, 1, 3, 4, 12, 22, 61, 128, 335, 756, 1936, 4580, 11652, 28402, 72209, 179460, 457274, 1151725, 2945129, 7489680, 19228598, 49256157, 126958030, 327072560, 846173899, 2190012371, 5685200054, 14770728584, 38463268482, 100259225816 (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 Vincenzo Librandi, 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.: z^2*A^2/(1-z*A), where A = (1+2*z+sqrt(1-4*z+4*z^2-4*z^4) -sqrt(2)*sqrt(1-4*z^2-2*z^4+(2*z+1)*sqrt(1-4*z+4*z^2-4*z^4)))/(4*z^2). a(n) ~ c * (1+sqrt(3))^n / n^(3/2), where c = 4/sqrt(Pi*(27 + 17*sqrt(3) - sqrt(2*(730 + 929*sqrt(3))/3))) = 0.5480566813380593118... - Vaclav Kotesovec, Feb 29 2016 a(n) = Sum_{m=2..n} (m*Sum_{i=0..n-m }((Sum_{j=0..i+m }(binomial(i+m,j)*binomial(j,i-j)))*Sum_{k=0..n-i-m }((binomial(2*k+i+m-1,k)*Sum_{l=0..k}(binomial(k,l)*binomial(k-l,n-3*l-k-i-m)*(-1)^(n-l-k-m)))/(k+i+m)))). - Vladimir Kruchinin, Mar 06 2016 A(x) = x^2*A005220(x)^2/(1-x*A005220(x)). - Gheorghe Coserea, Jan 16 2017 MATHEMATICA a = (2*z + Sqrt[-4*z^4 + 4*z^2 - 4*z + 1] - Sqrt[2]*Sqrt[-2*z^4 - 4*z^2 + (2*z + 1)*Sqrt[-4*z^4 + 4*z^2 - 4*z + 1] + 1] + 1)/(4*z^2); gf = z^2*a^2/(1 - z*a); CoefficientList[Series[gf, {z, 0, 31}], z] (* Jean-François Alcover, Dec 21 2012, from g.f. *) PROG (Maxima) a(n):=sum(m*sum((sum(binomial(i+m, j)*binomial(j, i-j), j, 0, i+m))*sum((binomial(2*k+i+m-1, k)*sum(binomial(k, l)*binomial(k-l, n-3*l-k-i-m)*(-1)^(n-l-k-m), l, 0, k))/(k+i+m), k, 0, n-i-m), i, 0, n-m), m, 2, n); /* Vladimir Kruchinin, Mar 06 2016 */ CROSSREFS Sequence in context: A075220 A075221 A129922 * A243391 A000206 A240737 Adjacent sequences:  A005218 A005219 A005220 * A005222 A005223 A005224 KEYWORD nonn,easy,nice,walk AUTHOR EXTENSIONS More terms from Emeric Deutsch, Dec 17 2003 STATUS approved

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