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

0,8

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.

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).

Links

Table of n, a(n) for n=0..135.

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)!).

Example

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;
  ...

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 *)

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.

Sequence in context: A286096 A247622 A256037 * A066325 A099174 A137297

Adjacent sequences: A179895 A179896 A179897 * A179899 A179900 A179901

Keyword

nonn,tabl

Author

N. J. A. Sloane, Jan 30 2011

Status

approved