|
| |
|
|
A005221
|
|
Number of Dyck paths of knight moves.
(Formerly M2371)
|
|
0
|
|
|
|
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
|
J. Labelle and Y.-N. Yeh, Dyck paths of knight moves, Discrete Applied Math., 24 (1989), 213-221.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
|
|
|
LINKS
|
Table of n, a(n) for n=0..31.
|
|
|
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).
|
|
|
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.*)
|
|
|
CROSSREFS
|
Sequence in context: A075220 A075221 A129922 * A000206 A075223 A071332
Adjacent sequences: A005218 A005219 A005220 * A005222 A005223 A005224
|
|
|
KEYWORD
|
nonn,easy,nice
|
|
|
AUTHOR
|
N. J. A. Sloane.
|
|
|
EXTENSIONS
|
More terms from Emeric Deutsch, Dec 17 2003
|
|
|
STATUS
|
approved
|
| |
|
|
