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A179898 Triangle V(l,p) (l>=0, p=0..l) read by rows: see Formula for definition, see Comments for motivation. 5
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 (list; table; graph; refs; listen; history; text; internal format)
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
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
KEYWORD
nonn,tabl,changed
AUTHOR
N. J. A. Sloane, Jan 30 2011
STATUS
approved

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Last modified April 14 03:49 EDT 2024. Contains 371655 sequences. (Running on oeis4.)