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

0,2

Comments

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

Links

Table of n, a(n) for n=0..50.

Peter Bala, Notes on generalized Riordan arrays

Peter Bala, A 4-parameter family of embedded Riordan arrays

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)

Example

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

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 *)

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.

Sequence in context: A297898 A322384 A360088 * A266577 A143411 A096773

Adjacent sequences: A113136 A113137 A113138 * A113140 A113141 A113142

Keyword

easy,nonn,tabl

Author

Paul Barry, Oct 15 2005

Status

approved