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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). 8
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

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

Table of n, a(n) for n=0..54.

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.

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).

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 *)

CROSSREFS

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

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

Sequence in context: A238731 A124037 A126126 * A286784 A047908 A125847

Adjacent sequences:  A090282 A090283 A090284 * A090286 A090287 A090288

KEYWORD

easy,nonn,tabl

AUTHOR

Philippe Deléham, Jan 24 2004

STATUS

approved

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Last modified May 6 16:20 EDT 2021. Contains 343586 sequences. (Running on oeis4.)