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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-Fran├žois 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

N. J. A. Sloane

EXTENSIONS

More terms from Emeric Deutsch, Dec 17 2003

STATUS

approved

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Last modified May 28 02:53 EDT 2017. Contains 287211 sequences.