A113139 - OEIS

%I #28 Feb 19 2020 00:57:43

%S 1,3,1,13,5,1,63,25,7,1,321,129,41,9,1,1683,681,231,61,11,1,8989,3653,

%T 1289,377,85,13,1,48639,19825,7183,2241,575,113,15,1,265729,108545,

%U 40081,13073,3649,833,145,17,1,1462563,598417,224143,75517,22363,5641

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

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

%H Peter Bala, <a href="/A260492/a260492.pdf">Notes on generalized Riordan arrays</a>

%H Peter Bala, <a href="/A264772/a264772_1.pdf">A 4-parameter family of embedded Riordan arrays</a>

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

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

%F From _Peter Bala_, Dec 09 2015: (Start)

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

%F 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 + ....

%F 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.

%F 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)

%F T(n,k) = (-1)^(n-k)*hypergeom([n+1, -n+k], [1], 2). - _Peter Luschny_, Mar 02 2017

%F From _Peter Bala_, Feb 16 2020: (Start)

%F 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.

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

%F 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)

%e Triangle begins

%e      1;

%e      3,    1;

%e     13,    5,    1;

%e     63,   25,    7,   1;

%e    321,  129,   41,   9,  1;

%e   1683,  681,  231,  61, 11,  1;

%e   8989, 3653, 1289, 377, 85, 13, 1;

%e   ...

%e A113139 as a square array = A110171 * A008288:

%e   / 1   1   1   1 ... \   / 1         \ / 1 1  1  1 ...\

%e   | 3   5   7   9 ... |   | 2  1       || 1 3  5  7 ...|

%e   |13  25  41  61 ... | = | 8  4 1     || 1 5 13 25 ...|

%e   |63 129 231 377 ... |   |38 18 6 1   || 1 7 25 63 .. |

%e   |...                |   |...         || 1...         |

%e - _Peter Bala_, Dec 09 2015

%p T := (n,k) -> (-1)^(n-k)*hypergeom([n+1, -n+k], [1], 2):

%p seq(seq(simplify(T(n,k)),k=0..n),n=0..8); # _Peter Luschny_, Mar 02 2017

%t 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 *)

%Y 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.

%K easy,nonn,tabl

%O 0,2

%A _Paul Barry_, Oct 15 2005