A267192 Column 3 of triangle in A059317 (the Pascal "Rhombus").

0, 0, 0, 1, 4, 19, 70, 261, 914, 3177, 10816, 36566, 122552, 408840, 1358032, 4497995, 14862112, 49019688, 161449208, 531152855, 1745892452, 5734722698, 18826352472, 61777432510, 202648614072, 664569581090, 2178948104572, 7143067052707, 23413795288008

Offset: 0

Author

N. J. A. Sloane, Jan 22 2016

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• José L. Ramírez, The Pascal Rhombus and the Generalized Grand Motzkin Paths, arXiv:1511.04577 [math.CO], 2015.

Crossrefs

Cf. A059317, A106050, A106053.

Programs

Maple

T:= proc(n, k) option remember; `if`(min(n, k)<0, 0,
      `if`(k=0, 1, T(n-1, k)+T(n-1, k-1)+T(n-1, k-2)+T(n-2, k-2)))
    end:
a:= n-> T(n, n-3):
seq(a(n), n=0..30);  # Alois P. Heinz, Jan 24 2016

Mathematica

T[n_, k_] := T[n, k] = If[Min[n, k]<0, 0, If[k == 0, 1, T[n-1, k] + T[n-1, k-1] + T[n-1, k-2] + T[n-2, k-2]]];
a[n_] := T[n, n-3];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jun 23 2017, after Alois P. Heinz *)

Formula

Conjecture: +(n-2)*(n-3)*(n+3)*a(n) -n*(2*n-1)*(n-2)*a(n-1) -(n-1)*(5*n^2-10*n+18)*a(n-2) +n*(2*n-3)*(n-2)*a(n-3) +n*(n+1)*(n-5)*a(n-4)=0. - R. J. Mathar, Jul 23 2017

Extensions

More terms from Alois P. Heinz, Jan 24 2016