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A357654 Number of 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 A357654 #14 Nov 08 2022 01:47:04
%S A357654 1,0,1,1,1,2,3,3,6,9,10,19,29,34,63,97,118,215,333,416,749,1165,1485,
%T A357654 2650,4135,5355,9490,14845,19473,34318,53791,71313,125104,196417,
%U A357654 262735,459152,721887,973027,1694914,2667941,3619955,6287896,9907851,13521307,23429158
%N A357654 Number of 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 A357654 Alois P. Heinz, <a href="/A357654/b357654.txt">Table of n, a(n) for n = 0..2500</a>
%H A357654 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%F A357654 a(n) = Sum_{k=0..floor(n/2)} A120730(n-k, k). - _G. C. Greubel_, Nov 07 2022
%p A357654 b:= proc(x, y) option remember; `if`(min(x, y)<0 or y>x, 0,
%p A357654       `if`(max(x, y)=0, 1, b(x-1, y)+b(x, y-1)))
%p A357654     end:
%p A357654 a:= n-> add(b(i, n-2*i), i=ceil(n/3)..floor(n/2)):
%p A357654 seq(a(n), n=0..44);
%t A357654 A120730[n_, k_]:= If[n>2*k, 0, Binomial[n,k]*(2*k-n+1)/(k+1)];
%t A357654 A357654[n_]:= Sum[A120730[n-k,k], {k,0,Floor[n/2]}];
%t A357654 Table[A357654[n], {n,0,50}] (* _G. C. Greubel_, Nov 07 2022 *)
%o A357654 (Magma)
%o A357654 A120730:= func< n, k | n gt 2*k select 0 else Binomial(n, k)*(2*k-n+1)/(k+1) >;
%o A357654 A357654:= func< n | (&+[A120730(n-k, k): k in [0..Floor(n/2)]]) >;
%o A357654 [A357654(n): n in [0..50]]; // _G. C. Greubel_, Nov 07 2022
%o A357654 (SageMath)
%o A357654 def A120730(n, k): return 0 if (n>2*k) else binomial(n, k)*(2*k-n+1)/(k+1)
%o A357654 def A357654(n): return sum(A120730(n-k,k) for k in range((n//2)+1))
%o A357654 [A357654(n) for n in range(51)] # _G. C. Greubel_, Nov 07 2022
%Y A357654 Cf. A120730, A165407, A357655.
%K A357654 nonn,walk
%O A357654 0,6
%A A357654 _Alois P. Heinz_, Oct 07 2022