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A283667 Number of Motzkin prefixes of length 2n and height n.

%I A283667 #13 May 11 2020 06:58:46
%S A283667 1,3,13,64,334,1802,9933,55575,314362,1793126,10295625,59430043,
%T A283667 344559826,2005026610,11703965955,68503652100,401892122682,
%U A283667 2362629703214,13914547415998,82081163986020,484893156220356,2868234297838092,16986185485228431,100703275233924096
%N A283667 Number of Motzkin prefixes of length 2n and height n.
%H A283667 Alois P. Heinz, <a href="/A283667/b283667.txt">Table of n, a(n) for n = 0..1279</a>
%F A283667 Recursion: see Maple program.
%F A283667 a(n) = A283595(2n,n).
%F A283667 a(n) ~ sqrt(769 + 2762/sqrt(13)) * (70 + 2*13^(3/2))^n / (3^(3*n+3)*sqrt(3*Pi*n)). - _Vaclav Kotesovec_, Mar 13 2017
%p A283667 a:= proc(n) option remember; `if`(n<2, 1+2*n, ((4*n-2)*(455*n^6-
%p A283667        1155*n^5-2776*n^4+1047*n^3+1493*n^2-72*n-72)*a(n-1)+36*
%p A283667       (n-1)*(2*n-1)*(2*n-3)*(13*n^4-7*n^3-64*n^2-44*n-6)*a(n-2))/
%p A283667       ((9*n+6)*(3*n+1)*(n+1)*(13*n^4-59*n^3+35*n^2+11*n-6)))
%p A283667     end:
%p A283667 seq(a(n), n=0..30);
%t A283667 b[x_, y_, m_] := b[x, y, m] = If[x == 0, z^m, b[x - 1, y, m] + If[y > 0, b[x - 1, y - 1, m], 0] + b[x - 1, y + 1, Max[m, y + 1]]];
%t A283667 a[n_] := SeriesCoefficient[b[2n, 0, 0], {z, 0, n}];
%t A283667 a /@ Range[0, 30] (* _Jean-François Alcover_, May 11 2020, after _Alois P. Heinz_ in A283595 *)
%Y A283667 Cf. A283595, A283799.
%K A283667 nonn
%O A283667 0,2
%A A283667 _Alois P. Heinz_, Mar 13 2017