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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 sub-diagonal 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-Fran├žois 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 * A099174 A066325 A137297

Adjacent sequences:  A179895 A179896 A179897 * A179899 A179900 A179901

KEYWORD

nonn,tabl

AUTHOR

N. J. A. Sloane, Jan 30 2011

STATUS

approved

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Last modified March 18 22:11 EDT 2019. Contains 321305 sequences. (Running on oeis4.)