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A286761 Total number of nodes summed over all lattice paths from (0,0) to (n,0) that do not go below the x-axis or above the diagonal x=y and consist of steps D=(1,-1), H=(1,0) and S=(0,1).

%I A286761 #9 Jun 28 2022 03:16:07
%S A286761 1,2,7,25,106,470,2218,10799,54158,277089,1441956,7602630,40524952,
%T A286761 217954222,1181107568,6441519814,35323986620,194629681327,
%U A286761 1076819450324,5979314763974,33308210757892,186074808452033,1042146006514656,5850075202736100,32907053660222560
%N A286761 Total number of nodes summed over all lattice paths from (0,0) to (n,0) that do not go below the x-axis or above the diagonal x=y and consist of steps D=(1,-1), H=(1,0) and S=(0,1).
%H A286761 Alois P. Heinz, <a href="/A286761/b286761.txt">Table of n, a(n) for n = 0..1000</a>
%p A286761 b:= proc(x, y) option remember; `if`(y<0 or y>x, 0, `if`(x=0, [1$2],
%p A286761       (p-> p+[0, p[1]])(b(x-1, y)+b(x, y-1)+b(x-1, y+1))))
%p A286761     end:
%p A286761 a:= n-> b(n, 0)[2]:
%p A286761 seq(a(n), n=0..30);
%t A286761 b[x_, y_] := b[x, y] = If[y < 0 || y > x, 0, If[x == 0, {1, 1}, Function[
%t A286761      p, p + {0, p[[1]]}][b[x - 1, y] + b[x, y - 1] + b[x - 1, y + 1]]]];
%t A286761 a[n_] := b[n, 0][[2]];
%t A286761 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Jun 28 2022, after _Alois P. Heinz_ *)
%Y A286761 Cf. A114296.
%K A286761 nonn
%O A286761 0,2
%A A286761 _Alois P. Heinz_, May 14 2017