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A357655 Total number of nodes summed over all lattice paths from (0,0) to (i,n-2*i) that do not go above the diagonal x=y using steps in {(1,0), (0,1)}.

%I A357655 #13 May 27 2023 09:13:13
%S A357655 1,0,2,3,3,8,14,15,35,59,69,147,245,300,608,1005,1269,2489,4091,5280,
%T A357655 10120,16565,21735,40950,66820,88815,165125,268785,361005,664108,
%U A357655 1078904,1461609,2665617,4323643,5899917,10682712,17304516,23759955,42759385,69187281
%N A357655 Total number of nodes summed over all lattice paths from (0,0) to (i,n-2*i) that do not go above the diagonal x=y using steps in {(1,0), (0,1)}.
%H A357655 Alois P. Heinz, <a href="/A357655/b357655.txt">Table of n, a(n) for n = 0..2500</a>
%H A357655 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%p A357655 b:= proc(x, y) option remember; `if`(min(x, y)<0 or y>x, 0, `if`(
%p A357655        max(x, y)=0, [1$2], (p-> p+[0, p[1]])(b(x-1, y)+b(x, y-1))))
%p A357655     end:
%p A357655 a:= n-> add(b(i, n-2*i), i=ceil(n/3)..floor(n/2))[2]:
%p A357655 seq(a(n), n=0..39);
%t A357655 b[x_, y_] := b[x, y] = If[Min[x, y] < 0 || y > x, {0, 0}, If[Max[x, y] == 0, {1, 1}, Function[p, p + {0, p[[1]]}][b[x - 1, y] + b[x, y - 1]]]];
%t A357655 a[n_] :=  If[n == 1, 0, Sum[b[i, n - 2i], {i, Ceiling[n/3], Floor[n/2]}][[2]]];
%t A357655 Table[a[n], {n, 0, 39}] (* _Jean-François Alcover_, May 27 2023, after _Alois P. Heinz_ *)
%Y A357655 Cf. A357654.
%K A357655 nonn,walk
%O A357655 0,3
%A A357655 _Alois P. Heinz_, Oct 07 2022