A113139 Number triangle, equal to half of Delannoy square array A008288.

1, 3, 1, 13, 5, 1, 63, 25, 7, 1, 321, 129, 41, 9, 1, 1683, 681, 231, 61, 11, 1, 8989, 3653, 1289, 377, 85, 13, 1, 48639, 19825, 7183, 2241, 575, 113, 15, 1, 265729, 108545, 40081, 13073, 3649, 833, 145, 17, 1, 1462563, 598417, 224143, 75517, 22363, 5641

Offset: 0

Author

Paul Barry, Oct 15 2005

Keywords

• easy
• nonn
• tabl

Comments

Row sums are A047781(n+1). Diagonal sums are A113140. Inverse is A113141.

Examples

			Triangle begins
     1;
     3,    1;
    13,    5,    1;
    63,   25,    7,   1;
   321,  129,   41,   9,  1;
  1683,  681,  231,  61, 11,  1;
  8989, 3653, 1289, 377, 85, 13, 1;
  ...
A113139 as a square array = A110171 * A008288:
  / 1   1   1   1 ... \   / 1         \ / 1 1  1  1 ...\
  | 3   5   7   9 ... |   | 2  1       || 1 3  5  7 ...|
  |13  25  41  61 ... | = | 8  4 1     || 1 5 13 25 ...|
  |63 129 231 377 ... |   |38 18 6 1   || 1 7 25 63 .. |
  |...                |   |...         || 1...         |
- _Peter Bala_, Dec 09 2015
		

Links

• Peter Bala, Notes on generalized Riordan arrays
• Peter Bala, A 4-parameter family of embedded Riordan arrays

Crossrefs

A001850 (column 0), A002002 (column 1), A026002 (column 2), A190666 (column 3), A047781 (row sums), A113140 (diagonal sums), A113141 (matrix inverse). Cf. A006318, A008288, A110171.

Programs

Maple

T := (n,k) -> (-1)^(n-k)*hypergeom([n+1, -n+k], [1], 2):
seq(seq(simplify(T(n,k)),k=0..n),n=0..8); # Peter Luschny, Mar 02 2017

Mathematica

Table[Sum[Binomial[n - k, j] Binomial[n + j, k + j], {j, 0, n}], {n, 0, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, Dec 09 2015 *)

Formula

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

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

From Peter Bala, Dec 09 2015: (Start)

T(n,k) = A008288(n - k, n).

O.g.f.: 2/( sqrt(x^2 - 6*x + 1)*(t*sqrt(x^2 - 6*x + 1) + t*x - t + 2) ) = 1 + (3 + t)*x + (13 + 5*t + t^2)*x^2 + ....

Riordan array (f(x), x*g(x)), where f(x) = 1/sqrt(1 - 6*x + x^2) is the o.g.f. for the central Delannoy numbers, A001850, and g(x) = 1/x* revert( x*(1 - x)/(1 + x) ) = 1 + 2*x + 6*x^2 + 22*x^3 + 90*x^4 + 394*x^5 + ... is the o.g.f. for the large Schroder numbers, A006318.

Read as a square array, this is the generalized Riordan array (f(x), g(x)) in the sense of the Bala link, which factorizes as (1 + x*g'(x)/g(x), x*g(x)) * (1/(1 - x), (1 + x)/(1 - x)) = A110171 * A008288. See the example below. (End)

T(n,k) = (-1)^(n-k)*hypergeom([n+1, -n+k], [1], 2). - Peter Luschny, Mar 02 2017

From Peter Bala, Feb 16 2020: (Start)

T(n,k) = P(n-k, k, 0, 3), where P(n, alpha, beta, x) is the n-th Jacobi polynomial with parameters alpha and beta.

T(n,k) = binomial(n,k) * hypergeom( [n + 1, k - n], [k + 1], -1 ).

The n-th row polynomial in descending powers of x is the n-th Taylor polynomial of the rational function (1 + x)^n/(1 - x)^(n+1) about 0. For example, for n = 4, (1 + x)^4/(1 - x)^5 = 1 + 9*x + 41*x^2 + 129*x^3 + 321*x^4 + O(x^5). Cf. A110171. (End)