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%I #33 Jun 12 2019 11:17:13
%S 1,1,1,2,4,1,5,15,7,1,14,56,37,10,1,42,210,176,68,13,1,132,792,794,
%T 392,108,16,1,429,3003,3473,2063,731,157,19,1,1430,11440,14893,10254,
%U 4395,1220,215,22,1,4862,43758,63004,49024,24465,8249,1886,282,25,1
%N Triangle T(n,k), 0<=k<=n, read by rows, defined by: T(n,k)=0 if k>n, T(n,0) = A000108(n); T(n+1,k)= Sum_{j=0..n} T(n-j,k-1)*binomial(2j+1,j+1).
%C The matrix inverse starts
%C 1;
%C -1, 1;
%C 2, -4, 1;
%C -4, 13, -7, 1;
%C 8, -38, 33, -10, 1;
%C -16, 104, -129, 62, -13, 1;
%C 32, -272, 450, -304, 100, -16, 1;
%C -64, 688, -1452, 1289, -590, 147, -19, 1;
%C 128, -1696, 4424, -4942, 2945, -1014, 203, -22, 1;
%C - _R. J. Mathar_, Mar 15 2013
%C Riordan array (c(x), x*c(x)^2/(1-x*c(x)^2)) where c(x) is the g.f. for the Catalan numbers (A000108). - _Philippe Deléham_, Jun 02 2013
%C The matrix inverse is the Riordan array ((1+x)/(1+2*x), x*(1+x)/(1+2*x)^2). - _Philippe Deléham_, Jan 26 2014
%H Pudwell, Lara; Scholten, Connor; Schrock, Tyler; Serrato, Alexa <a href="https://doi.org/10.1155/2014/316535">Noncontiguous pattern containment in binary trees</a>, ISRN Comb. 2014, Article ID 316535, 8 p. (2014), Table 1.
%H Efrat Engel Shaposhnik, <a href="http://hdl.handle.net/1721.1/104603">Antichains of Interval Orders and Semiorders, and Dilworth Lattices of maximum size Antichains</a>, Massachusetts Institute of Technology, June 2016.
%F T(n, 1) = n*A000108(n) = A001791(n) .
%F T(n, 2) = 2^(2n-1) - binomial(2n+1, n) + binomial(2n-1, n-1) = A006419(n).
%p A090285 := proc(n,k)
%p if k < 0 or k > n then
%p 0 ;
%p elif k = 0 then
%p A000108(n)
%p else
%p add(procname(n-1-j,k-1)*binomial(2*j+1,j+1),j=0..n-1) ;
%p end if;
%p end proc: # _R. J. Mathar_, Mar 15 2013
%t T[n_, k_] := T[n, k] = If[k == 0, CatalanNumber@ n, Sum[T[(n - 1) - j, k - 1] Binomial[2 j + 1, j + 1], {j, 0, n - 1}]]; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Michael De Vlieger_, Jun 26 2017 *)
%Y Diagonals: A000108, A001791, A006419; A000012, A016777.
%Y See also A001700 for binomial(2n+1,n+1).
%K easy,nonn,tabl
%O 0,4
%A _Philippe Deléham_, Jan 24 2004
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