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A005222 Number of Dyck paths of knight moves.

%I A005222 M3234 #28 Dec 26 2021 21:05:46
%S A005222 1,0,1,0,4,4,18,26,86,158,462,976,2665,6082,16040,38338,99536,244880,
%T A005222 631923,1583796,4081939,10358670,26728731,68425494,176964795,
%U A005222 455967376,1182454137,3061954102,7962768190,20702327552,53983118006,140817757006
%N A005222 Number of Dyck paths of knight moves.
%D A005222 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005222 Vaclav Kotesovec, <a href="/A005222/b005222.txt">Table of n, a(n) for n = 0..1000</a>
%H A005222 Vaclav Kotesovec, <a href="/A005222/a005222.txt">Recurrence (of order 11)</a>
%H A005222 J. Labelle and Y.-N. Yeh, <a href="http://dx.doi.org/10.1016/0166-218X(92)90286-J">Dyck paths of knight moves</a>, Discrete Applied Math., 24 (1989), 213-221.
%F A005222 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].
%F A005222 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
%F A005222 A(x) = x^2*A005220(x)*A005221(x) + x*A005221(x)^2 + A005220(x). - _Gheorghe Coserea_, Jan 16 2017
%t A005222 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)];
%t A005222 A[x] + O[x]^32 // CoefficientList[#, x]& (* _Jean-François Alcover_, Mar 26 2017, after _Gheorghe Coserea_ *)
%K A005222 nonn,easy,nice,walk
%O A005222 0,5
%A A005222 _N. J. A. Sloane_
%E A005222 More terms from _Emeric Deutsch_, Dec 17 2003