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A088457 Number of single nodes (exactly one node on that level) for all Motzkin paths of length n.

%I A088457 #27 Apr 16 2024 18:00:10
%S A088457 1,0,1,2,4,8,18,44,113,296,782,2076,5538,14856,40100,108936,297793,
%T A088457 818832,2263481,6286498,17532707,49077268,137821247,388150322,
%U A088457 1095980561,3101840232,8797579789,25001305410,71179961918,203000438544,579876376729,1658948939262
%N A088457 Number of single nodes (exactly one node on that level) for all Motzkin paths of length n.
%C A088457 A Motzkin path of length n is a sequence [y(0),...,y(n)] such that |y(i)-y(i+1)| <= 1, 0=y(0)=y(n)<=y(i).
%H A088457 Alois P. Heinz, <a href="/A088457/b088457.txt">Table of n, a(n) for n = 0..750</a>
%e A088457 [0,0,0,1,0], [0,0,1,0,0], [0,1,0,0,0], [0,1,2,1,0] are the a(4) = 4 sequences.
%p A088457 b:= proc(x, y, h, c) option remember; `if`(y<0 or y>x, 0,
%p A088457      `if`(x=0, c, add(b(x-1, y-i, max(h, y),
%p A088457      `if`(h=y, 0, `if`(h<y, 1, c))), i=-1..1)))
%p A088457     end:
%p A088457 a:= n-> b(n, 0$2, 1):
%p A088457 seq(a(n), n=0..31);  # _Alois P. Heinz_, Jul 25 2023
%t A088457 b[x_, y_, h_, c_] := b[x, y, h, c] = If[y<0 || y>x, 0, If[x == 0, c, Sum[b[x-1, y-i, Max[h, y], If[h == y, 0, If[h < y, 1, c]]], {i, -1, 1}]]];
%t A088457 a[n_] := b[n, 0, 0, 1];
%t A088457 Table[a[n], {n, 0, 31}] (* _Jean-François Alcover_, Oct 23 2023, after _Alois P. Heinz_ *)
%o A088457 (PARI) {a(n)=local(p0, p1, p2); if(n<0, 0, p1=1; polcoeff(sum(i=0, n, if(p2=(1-x)*p1-x^2*p0, p0=p1; p1=p2; (x^i/p0)^2), x*O(x^n)), n))}
%Y A088457 Cf. A001006, A051485, A152880.
%Y A088457 Column k=1 of A364386 and of A372014.
%K A088457 nonn
%O A088457 0,4
%A A088457 _Michael Somos_, Oct 01 2003
%E A088457 a(30)-a(31) from _Alois P. Heinz_, Jul 21 2023