A113955 Riordan array (1/((1-4*x)*c(x)),x*c(x)/sqrt(1-4*x)), c(x) the g.f. of A000108.

1, 3, 1, 11, 6, 1, 42, 30, 9, 1, 163, 140, 58, 12, 1, 638, 630, 325, 95, 15, 1, 2510, 2772, 1686, 624, 141, 18, 1, 9908, 12012, 8330, 3682, 1064, 196, 21, 1, 39203, 51480, 39796, 20264, 7050, 1672, 260, 24, 1, 155382, 218790, 185517, 106203, 42849, 12303, 2475, 333, 27, 1

Offset

0,2

Links

Table of n, a(n) for n=0..54.

Lili Mu, Yuanyuan Xing, and Sai-Nan Zheng, A New Criterion for the Total Positivity of Riordan Arrays, Journal of Integer Sequences, Vol. 28 (2025), Article 25.7.5. See p. 3.

Formula

Riordan array ((1/(1-4*x)+1/sqrt(1-4*x))/2, (2*x/((1-4*x)+sqrt(1-4*x)))).

Number triangle T(n, k) = Sum_{j=0..n} C(j, j-k)*C(2*n, n-j).

T(n,k) = Sum_{j=0..n} C(2*n,j)*C(n-j,k). - Paul Barry, Apr 03 2006

Example

Triangle begins:
    1;
    3,   1;
   11,   6,   1;
   42,  30,   9,  1;
  163, 140,  58, 12,  1;
  638, 630, 325, 95, 15, 1;
  ...

Crossrefs

Columns include A032443, A002457, A018218, A038836. Row sums are A100192. Diagonal sums are A113956.

Sequence in context: A388733 A092808 A343171 * A110165 A111965 A110440

Adjacent sequences: A113952 A113953 A113954 * A113956 A113957 A113958

Keyword

easy,nonn,tabl

Author

Paul Barry, Nov 09 2005

Extensions

More terms from Stefano Spezia, Dec 18 2025

Status

approved