

A158483


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


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
(list;
graph;
refs;
listen;
history;
text;
internal format)



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 (n2*k1,n+2*k+1).
2) The entries in column k are (with an offset of 2*k+1) the number of nth 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(14*x)^(1/2)]/(2*x) is the o.g.f. for the Catalan numbers A000108.
Row sums A000100.


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

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



