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%I #24 Mar 31 2024 17:29:15
%S 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,
%T 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,
%U 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
%N Triangle V(l,p) (l>=0, p=0..l) read by rows: see Formula for definition, see Comments for motivation.
%C 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.
%D 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).
%F 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)!).
%e Triangle begins:
%e 1;
%e 0, 1;
%e 1, 0, 1;
%e 0, 3, 0, 1;
%e 3, 0, 6, 0, 1;
%e 0, 14, 0, 10, 0, 1;
%e 14, 0, 40, 0, 15, 0, 1;
%e 0, 84, 0, 90, 0, 21, 0, 1;
%e 84, 0, 300, 0, 175, 0, 28, 0, 1;
%e 0, 594, 0, 825, 0, 308, 0, 36, 0, 1;
%e 594, 0, 2475, 0, 1925, 0, 504, 0, 45, 0, 1;
%e 0, 4719, 0, 7865, 0, 4004, 0, 780, 0, 55, 0, 1;
%e 4719, 0, 22022, 0, 21021, 0, 7644, 0, 1155, 0, 66, 0, 1;
%e 0, 40898, 0, 78078, 0, 49686, 0, 13650, 0, 1650, 0, 78, 0, 1;
%e 40898, 0, 208208, 0, 231868, 0, 107016, 0, 23100, 0, 2288, 0, 91, 0, 1;
%e ...
%p V:=proc(l,p)
%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;
%p end;
%p r:=n->[seq( V(n,p),p=0..n)];
%p for n from 0 to 15 do lprint(r(n)); od:
%t 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 *)
%Y The first two columns are both A005700, the next column is A181571. The diagonals on the right give A000217, A005701. Row sums are A005817.
%K nonn,tabl
%O 0,8
%A _N. J. A. Sloane_, Jan 30 2011
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