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A293944 Triangle read by rows related to Catalan triangle A009766.

%I A293944 #11 Apr 07 2020 08:47:52
%S A293944 1,1,1,2,3,2,5,9,9,5,14,28,34,28,14,42,90,123,123,90,42,132,297,440,
%T A293944 497,440,297,132,429,1001,1573,1935,1935,1573,1001,429,1430,3432,5642,
%U A293944 7397,8068,7397,5642,3432,1430,4862,11934,20332,28014,32636,32636,28014,20332,11934
%N A293944 Triangle read by rows related to Catalan triangle A009766.
%H A293944 Laurent Méhats, Lutz Straßburger, <a href="https://doi.org/10.1007/978-3-662-53826-5_13">Non-crossing Tree Realizations of Ordered Degree Sequences</a>, 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).
%e A293944 Triangle begins:
%e A293944 1,
%e A293944 1,1,
%e A293944 2,3,2,
%e A293944 5,9,9,5,
%e A293944 14,28,34,28,14,
%e A293944 42,90,123,123,90,42,
%e A293944 132,297,440,497,440,297,132,
%e A293944 ...
%p A293944 A000108 := proc(q)
%p A293944     if q <0 then
%p A293944         0;
%p A293944     else
%p A293944         binomial(2*q,q)/(1+q) ;
%p A293944     end if;
%p A293944 end proc:
%p A293944 R := proc(q,s)
%p A293944     option remember;
%p A293944     local a,j,l ;
%p A293944     if q= 0 then
%p A293944         A000108(s) ;
%p A293944     elif s = 0 then
%p A293944         A000108(q) ;
%p A293944     else
%p A293944         a := 0 ;
%p A293944         for j from 0 to q do
%p A293944             for l from 0 to s do
%p A293944                 if j+l-1 >= 0 then
%p A293944                     a := a+A000108(j+l-1) *procname(q-j,s-l) ;
%p A293944                 end if;
%p A293944             end do:
%p A293944         end do:
%p A293944     end if;
%p A293944 end proc:
%p A293944 A293944 := proc(n,k)
%p A293944     R(n-k,k) ;
%p A293944 end proc:
%p A293944 seq(seq(A293944(n,k),k=0..n),n=0..12) ; # _R. J. Mathar_, Nov 02 2017
%t A293944 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 A293944 T [n_, k_] := R[n - k, k];
%t A293944 Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Apr 07 2020, after _R. J. Mathar_ *)
%Y A293944 Cf. A009766, A000108 (1st column), A000245 (2nd column), A120989 (3rd), A090317 (row sums).
%K A293944 nonn,tabl
%O A293944 0,4
%A A293944 _N. J. A. Sloane_, Oct 21 2017
%E A293944 More terms from _R. J. Mathar_, Nov 02 2017