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

0,3

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].

Links

Table of n, a(n) for n=0..46.

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

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.

Example

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

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;

Crossrefs

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

Sequence in context: A304249 A128727 A126177 * A128733 A128724 A128753

Adjacent sequences: A158480 A158481 A158482 * A158484 A158485 A158486

Keyword

easy,nonn,tabf

Author

Peter Bala, Mar 20 2009

Status

approved