A293944 Triangle read by rows related to Catalan triangle A009766.

1, 1, 1, 2, 3, 2, 5, 9, 9, 5, 14, 28, 34, 28, 14, 42, 90, 123, 123, 90, 42, 132, 297, 440, 497, 440, 297, 132, 429, 1001, 1573, 1935, 1935, 1573, 1001, 429, 1430, 3432, 5642, 7397, 8068, 7397, 5642, 3432, 1430, 4862, 11934, 20332, 28014, 32636, 32636, 28014, 20332, 11934

Offset: 0

Author

N. J. A. Sloane, Oct 21 2017

Keywords

• nonn
• tabl

Examples

			Triangle begins:
1,
1,1,
2,3,2,
5,9,9,5,
14,28,34,28,14,
42,90,123,123,90,42,
132,297,440,497,440,297,132,
...
		

Links

• Laurent Méhats, Lutz Straßburger, Non-crossing Tree Realizations of Ordered Degree Sequences, Pages 211-227 in Logical Aspects of Computational Linguistics. Celebrating 20 Years of LACL (1996-2016), 9th International Conference, LACL 2016, Nancy, France, December 5-7, 2016, Proceedings, Lecture Notes in Computer Science book series (LNCS, volume 10054). See Eq. (7).

Crossrefs

Cf. A009766, A000108 (1st column), A000245 (2nd column), A120989 (3rd), A090317 (row sums).

Programs

Maple

A000108 := proc(q)
    if q <0 then
        0;
    else
        binomial(2*q,q)/(1+q) ;
    end if;
end proc:
R := proc(q,s)
    option remember;
    local a,j,l ;
    if q= 0 then
        A000108(s) ;
    elif s = 0 then
        A000108(q) ;
    else
        a := 0 ;
        for j from 0 to q do
            for l from 0 to s do
                if j+l-1 >= 0 then
                    a := a+A000108(j+l-1) *procname(q-j,s-l) ;
                end if;
            end do:
        end do:
    end if;
end proc:
A293944 := proc(n,k)
    R(n-k,k) ;
end proc:
seq(seq(A293944(n,k),k=0..n),n=0..12) ; # R. J. Mathar, Nov 02 2017

Mathematica

R[q_, s_] := R[q, s] = Module[{a, j, l}, If[q == 0, CatalanNumber[s], If[s == 0, CatalanNumber[q], a = 0; For[j = 0, j <= q, j++, For[l = 0, l <= s , l++, If[j + l - 1 >= 0, a = a + CatalanNumber[j + l - 1] R[q - j, s - l]] ]]]] /. Null -> a];
T [n_, k_] := R[n - k, k];
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 07 2020, after R. J. Mathar *)

Extensions

More terms from R. J. Mathar, Nov 02 2017