A286918 - OEIS

%I #19 Dec 29 2020 09:04:13

%S 1,3,21,159,1257,10046,81811,674184,5605141,46920874,394949193,

%T 3339464105,28343082002,241324470723,2060357315568,17632454524499,

%U 151211115930880,1299121121946684,11179500360964780,96344672766997340,831385476662968094,7182777882757416692

%N 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 Alois P. Heinz, <a href="/A286918/b286918.txt">Table of n, a(n) for n = 0..1000</a>

%F 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 b:= proc(x, y) option remember; `if`(y>x or y<0, 0,

%p       `if`(x=0, [1$2], add((p-> p+[0, p[1]])(

%p          b(x-j, y)+b(x, y-j)), j=1..4)))

%p     end:

%p a:= n-> b(n$2)[2]:

%p seq(a(n), n=0..25);

%t 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 a[n_] := b[n, n][[2]];

%t a /@ Range[0, 25] (* _Jean-François Alcover_, Dec 29 2020, after _Alois P. Heinz_ *)

%Y Cf. A175891.

%K nonn

%O 0,2

%A _Alois P. Heinz_, May 17 2017