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A286918 Total number of nodes summed over all lattice paths from (0,0) to (n,n) using steps {(k,0), (0,k) | 0 which never go above the diagonal x=y.

%I A286918 #19 Dec 29 2020 09:04:13
%S A286918 1,3,21,159,1257,10046,81811,674184,5605141,46920874,394949193,
%T A286918 3339464105,28343082002,241324470723,2060357315568,17632454524499,
%U A286918 151211115930880,1299121121946684,11179500360964780,96344672766997340,831385476662968094,7182777882757416692
%N A286918 Total number of nodes summed over all lattice paths from (0,0) to (n,n) using steps {(k,0), (0,k) | 0<k<=4} which never go above the diagonal x=y.
%H A286918 Alois P. Heinz, <a href="/A286918/b286918.txt">Table of n, a(n) for n = 0..1000</a>
%F A286918 a(n) ~ c * d^n / sqrt(n), where d = 8.84734830841870961487278801886633962039798... is the real root of the equation 4 + 4*d - 8*d^2 - 8*d^3 + d^4 = 0 and c = 0.43633259077568249345422000202799136319250347607927734138960545201547... - _Vaclav Kotesovec_, May 30 2017
%p A286918 b:= proc(x, y) option remember; `if`(y>x or y<0, 0,
%p A286918       `if`(x=0, [1$2], add((p-> p+[0, p[1]])(
%p A286918          b(x-j, y)+b(x, y-j)), j=1..4)))
%p A286918     end:
%p A286918 a:= n-> b(n$2)[2]:
%p A286918 seq(a(n), n=0..25);
%t A286918 b[x_, y_] := b[x, y] = If[y > x || y < 0, {0, 0}, If[x == 0, {1, 1}, Sum[Function[p, p + {0, p[[1]]}][b[x-j, y] + b[x, y-j]], {j, 1, 4}]]];
%t A286918 a[n_] := b[n, n][[2]];
%t A286918 a /@ Range[0, 25] (* _Jean-François Alcover_, Dec 29 2020, after _Alois P. Heinz_ *)
%Y A286918 Cf. A175891.
%K A286918 nonn
%O A286918 0,2
%A A286918 _Alois P. Heinz_, May 17 2017