0,3

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].

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

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 A000100.

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

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;

A000245 (column 0), A000588 (column 1), A000589 (column 2), A00100 (row sums), A119245.

Sequence in context: A126179 A128727 A126177 this_sequence A128733 A128724 A128753

Adjacent sequences: A158480 A158481 A158482 this_sequence A158484 A158485 A158486

easy,nonn,tabf

Peter Bala (pbala(AT)talktalk.net), Mar 20 2009