A090285 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).

1, 1, 1, 2, 4, 1, 5, 15, 7, 1, 14, 56, 37, 10, 1, 42, 210, 176, 68, 13, 1, 132, 792, 794, 392, 108, 16, 1, 429, 3003, 3473, 2063, 731, 157, 19, 1, 1430, 11440, 14893, 10254, 4395, 1220, 215, 22, 1, 4862, 43758, 63004, 49024, 24465, 8249, 1886, 282, 25, 1

Offset: 0

Author

Philippe Deléham, Jan 24 2004

Keywords

• easy
• nonn
• tabl

Comments

The matrix inverse starts

1;

-1, 1;

2, -4, 1;

-4, 13, -7, 1;

8, -38, 33, -10, 1;

-16, 104, -129, 62, -13, 1;

32, -272, 450, -304, 100, -16, 1;

-64, 688, -1452, 1289, -590, 147, -19, 1;

128, -1696, 4424, -4942, 2945, -1014, 203, -22, 1;

- R. J. Mathar, Mar 15 2013

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

The matrix inverse is the Riordan array ((1+x)/(1+2*x), x*(1+x)/(1+2*x)^2). - Philippe Deléham, Jan 26 2014

Links

• Pudwell, Lara; Scholten, Connor; Schrock, Tyler; Serrato, Alexa Noncontiguous pattern containment in binary trees, ISRN Comb. 2014, Article ID 316535, 8 p. (2014), Table 1.
• Efrat Engel Shaposhnik, Antichains of Interval Orders and Semiorders, and Dilworth Lattices of maximum size Antichains, Massachusetts Institute of Technology, June 2016.

Crossrefs

Diagonals: A000108, A001791, A006419; A000012, A016777.

See also A001700 for binomial(2n+1,n+1).

Programs

Maple

A090285 := proc(n,k)
    if k < 0 or k > n then
        0 ;
    elif k = 0 then
        A000108(n)
    else
        add(procname(n-1-j,k-1)*binomial(2*j+1,j+1),j=0..n-1) ;
    end if;
end proc: # R. J. Mathar, Mar 15 2013

Mathematica

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 *)

Formula

T(n, 1) = n*A000108(n) = A001791(n) .

T(n, 2) = 2^(2n-1) - binomial(2n+1, n) + binomial(2n-1, n-1) = A006419(n).