A158483 Triangle read by rows: T(n,k) = (4k+3)/(n+2k+2)*binomial(2n,n+2k+1).

0, 1, 3, 9, 1, 28, 7, 90, 35, 1, 297, 154, 11, 1001, 637, 77, 1, 3432, 2548, 440, 15, 11934, 9996, 2244, 135, 1, 41990, 38760, 10659, 950, 19, 149226, 149226, 48279, 5775, 209, 1, 534888, 572033, 211508, 31878, 1748, 23, 1931540, 2187185, 904475, 164450

Offset: 0

Author

Peter Bala, Mar 20 2009

Keywords

• easy
• nonn
• tabf

Comments

This triangle forms a companion to A119245.

Combinatorial interpretations of T(n,k):

1) The number of standard tableaux of shape (n-2*k-1,n+2*k+1).

2) The entries in column k are (with an offset of 2*k+1) the number of n-th generation vertices in the tree of sequences with unit increase labeled by 4*k+2. See [Sunik, Theorem 4].

Examples

			Triangle begins
==================================
n\k|.....0.....1.....2.....3.....4
==================================
.0.|.....0
.1.|.....1
.2.|.....3
.3.|.....9.....1
.4.|....28.....7
.5.|....90....35.....1
.6.|...297...154....11
.7.|..1001...637....77.....1
.8.|..3432..2548...440....15
.9.|.11934..9996..2244...135.....1
		

Links

• Zoran Sunic, Self describing sequences and the Catalan family tree, Elect. J. Combin., 10 (No. 1, 2003).

Crossrefs

Cf. A000245 (column 0), A000588 (column 1), A000589 (column 2), A001700 (row sums), A119245.

Programs

Maple

with(combinat): T:=(n,k) -> (4k+3)/(n+2k+2)*binomial(2n,n+2k+1): for n from 0 to 13 do seq(T(n,k),k = 0..6); end do;

Formula

T(n,k) = (4*k+3)/(n+2*k+2)*binomial(2*n,n+2*k+1).

O.g.f. y*C(y)^3/(1 - x*y^2*C(y)^4) = y + 3*y^2 + (9 + x)*y^3 + (28 + 7*x)*y^4 + ..., where C(x) = [1-(1-4*x)^(1/2)]/(2*x) is the o.g.f. for the Catalan numbers A000108.

Row sums A001700.