%I #34 Jun 07 2023 11:12:59
%S 0,1,1,2,5,12,31,83,227,634,1799,5171,15027,44074,130299,387880,
%T 1161665,3497734,10581819,32150411,98057835,300116888,921456715,
%U 2837379238,8760199757,27112737988,84103586027,261435982873,814257033047
%N Expansion of (1-x-sqrt(1-2*x-3*x^2-4*x^3))/(2*x*(1+x)).
%C a(n) counts Horse permutations of length n-1 (see Hou and Mansour reference, Proposition 3.1). - _David Callan_, Aug 27 2014
%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Barry/barry601.html">On Motzkin-Schröder Paths, Riordan Arrays, and Somos-4 Sequences</a>, J. Int. Seq. (2023) Vol. 26, Art. 23.4.7.
%H D. Merlini, D. G. Rogers, R. Sprugnoli and M. C. Verri, <a href="http://dx.doi.org/10.4153/CJM-1997-015-x">On some alternative characterizations of Riordan arrays</a>, Canad. J. Math., 49 (1997), 301-320.
%H Q. Hou and T. Mansour, <a href="https://doi.org/10.1016/j.dam.2005.11.004">Horse paths, restricted 132-avoiding permutations, continued fractions, and Chebyshev polynomials</a>, Discrete Applied Mathematics 154:8 (2006) 1183-1197.
%F a(n+1) = sum(k=0..n/2, binomial(2*k,k)/(k+1) * sum(i=0..k, binomial(k,i)*binomial(n-i,2*k) ) ).
%F D-finite with recurrence: (for b(n)=a(n+1)): 0 = 2*(n^2+14*n+48)*b(n+6) + (n^2+11*n+24)*b(n+5) - 2*(7*n^2+74*n+198)*b(n+4) - 2*(14*n^2+133*n+309)*b(n+3) - 6*(4*n^2+33*n+66)*b(n+2) - (5*n^2+49*n+90)*b(n+1) + 2*(2*n^2+7*n+6)*b(n). [_Emanuele Munarini_, May 06 2011]
%F a(0)=0, a(1)=1, a(2)=1, a(3)=2, a(n) = ((n-2)*a(n-1) +(5*n-7)*a(n-2) +(7*n-20) *a(n-3) +(4*n-14)*a(n-4))/(n+1). - _Tani Akinari_, Jul 03 2013
%t Table[Sum[Binomial[2k,k]/(k+1)Sum[Binomial[k,i]Binomial[n-i,2k],{i,0,k}],{k,0,n/2}],{n,0,29}] (* for a(n+1) *) (* _Emanuele Munarini_, May 06 2011 *)
%o (Maxima) makelist(sum(binomial(2*k,k)/(k+1)*sum(binomial(k,i)*binomial(n-i,2*k),i,0,k),k,0,n/2),n,0,29); (* for a(n+1) *) [_Emanuele Munarini_, May 06 2011]
%K nonn
%O 0,4
%A _N. J. A. Sloane_, Jun 12 2002