A100247 Slanted Catalan convolution table, read by rows of 2*n+1 terms in row n, where T(n,k) = C(n+2*k-[k/2],k)*(n-[k/2])/(n+2*k-[k/2]).

1, 1, 1, 0, 1, 2, 2, 5, 0, 1, 3, 5, 14, 14, 42, 0, 1, 4, 9, 28, 42, 132, 132, 429, 0, 1, 5, 14, 48, 90, 297, 429, 1430, 1430, 4862, 0, 1, 6, 20, 75, 165, 572, 1001, 3432, 4862, 16796, 16796, 58786, 0, 1, 7, 27, 110, 275, 1001, 2002, 7072, 11934, 41990, 58786, 208012

Offset

0,6

Comments

Row sums form A100248. Antidiagonal sums form A100249.

Links

Table of n, a(n) for n=0..60.

Formula

T(n, k) = A033184(n-[k/2], k) for n>0 (with A033184 formatted as a square array).

G.f. A(x, y) satisfies:

A(x^2, y) = ( (1+x)/(2*y - x*(1 - sqrt(1 - 4*x*y))) - (1-x)/(2*y + x*(1 - sqrt(1 + 4*x*y))) )*y/x.

Example

Rows begin:
[1],
[1,1,0],
[1,2,2,5,0],
[1,3,5,14,14,42,0],
[1,4,9,28,42,132,132,429,0],
[1,5,14,48,90,297,429,1430,1430,4862,0],
[1,6,20,75,165,572,1001,3432,4862,16796,16796,58786,0],...
and is derived from the square array of Catalan convolutions (A033184)
by shifting each column k down by [k/2] rows.

Prog

(PARI) {T(n, k) = if(n==k&k==0, 1, binomial(n+2*k-(k\2), k)*(n-(k\2))/(n+2*k-(k\2)))}
for(n=0, 10, for(k=0, 2*n, print1(T(n, k), ", ")); print(""))
(PARI) {T(n, k) = polcoeff(((1-sqrt(1-4*z +z*O(z^(2*n))))/(2*z))^(n-k\2), k, z)}
for(n=0, 10, for(k=0, 2*n, print1(T(n, k), ", ")); print(""))

Crossrefs

Cf. A033184, A100248, A100249.

Sequence in context: A363842 A292383 A332896 * A194123 A011342 A084046

Adjacent sequences: A100244 A100245 A100246 * A100248 A100249 A100250

Keyword

nonn,tabf

Author

Paul D. Hanna, Nov 09 2004

Status

approved