A179898 Triangle V(l,p) (l>=0, p=0..l) read by rows: see Formula for definition, see Comments for motivation.

1, 0, 1, 1, 0, 1, 0, 3, 0, 1, 3, 0, 6, 0, 1, 0, 14, 0, 10, 0, 1, 14, 0, 40, 0, 15, 0, 1, 0, 84, 0, 90, 0, 21, 0, 1, 84, 0, 300, 0, 175, 0, 28, 0, 1, 0, 594, 0, 825, 0, 308, 0, 36, 0, 1, 594, 0, 2475, 0, 1925, 0, 504, 0, 45, 0, 1, 0, 4719, 0, 7865, 0, 4004, 0, 780, 0, 55, 0, 1, 4719, 0, 22022, 0, 21021, 0, 7644, 0, 1155, 0, 66, 0, 1, 0, 40898, 0, 78078, 0, 49686, 0, 13650, 0, 1650, 0, 78, 0, 1, 40898, 0, 208208, 0, 231868, 0, 107016, 0, 23100, 0, 2288, 0, 91, 0, 1, 0, 379236, 0, 804440, 0, 606424, 0, 214200, 0, 37400, 0, 3094, 0, 105, 0, 1

Offset: 0

Author

N. J. A. Sloane, Jan 30 2011

Keywords

• nonn
• tabl

Comments

If the triangle in A053121 is regarded as counting minimal subdiagonal paths in the first quadrant, this triangle enumerates pairs of non-crossing paths of the same type.

Examples

			Triangle begins:
  1;
  0, 1;
  1, 0, 1;
  0, 3, 0, 1;
  3, 0, 6, 0, 1;
  0, 14, 0, 10, 0, 1;
  14, 0, 40, 0, 15, 0, 1;
  0, 84, 0, 90, 0, 21, 0, 1;
  84, 0, 300, 0, 175, 0, 28, 0, 1;
  0, 594, 0, 825, 0, 308, 0, 36, 0, 1;
  594, 0, 2475, 0, 1925, 0, 504, 0, 45, 0, 1;
  0, 4719, 0, 7865, 0, 4004, 0, 780, 0, 55, 0, 1;
  4719, 0, 22022, 0, 21021, 0, 7644, 0, 1155, 0, 66, 0, 1;
  0, 40898, 0, 78078, 0, 49686, 0, 13650, 0, 1650, 0, 78, 0, 1;
  40898, 0, 208208, 0, 231868, 0, 107016, 0, 23100, 0, 2288, 0, 91, 0, 1;
  ...
		

References

• D. Gouyou-Beauchamps, Chemins sous-diagonaux et tableau de Young, pp. 112-125 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, 1986 (see |V_{l,p}| on page 114).

Crossrefs

The first two columns are both A005700, the next column is A181571. The diagonals on the right give A000217, A005701. Row sums are A005817.

Programs

Maple

V:=proc(l,p)
if ((l-p) mod 2) = 1 then 0 else l!*(l+2)!*(p+3)! / (((l-p)/2)!*((l-p)/2+1)!*p!*((l+p)/2+2)!*((l+p)/2+3)!); fi;
end;
r:=n->[seq( V(n,p),p=0..n)];
for n from 0 to 15 do lprint(r(n)); od:

Mathematica

v[l_, p_] := If[Mod[l-p, 2] == 1, 0, l!*(l+2)!*(p+3)!/(((l-p)/2)!*((l-p)/2+1)!*p!*((l+p)/2+2)!*((l+p)/2+3)!)]; Table[v[l, p], {l, 0, 15}, {p, 0, l}] // Flatten (* Jean-François Alcover, Jan 09 2014, translated from Maple *)

Formula

V(l,p) = 0 if l and p have opposite parity, otherwise V(l,p) = l!*(l+2)!*(p+3)!/(((l-p)/2)!*((l-p)/2+1)!*p!*((l+p)/2+2)!*((l+p)/2+3)!).