A110171 -id:A110171 - cp's OEIS frontend

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

Paul Barry, Oct 15 2005

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

			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

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

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.

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

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 *)

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)

A118384 Gaussian column reduction of Hankel matrix for central Delannoy numbers.

Original entry on oeis.org

1, 3, 1, 13, 6, 1, 63, 33, 9, 1, 321, 180, 62, 12, 1, 1683, 985, 390, 100, 15, 1, 8989, 5418, 2355, 720, 147, 18, 1, 48639, 29953, 13923, 4809, 1197, 203, 21, 1, 265729, 166344, 81340, 30744, 8806, 1848, 268, 24, 1, 1462563, 927441, 471852, 191184, 60858

Offset: 0

Paul Barry, Apr 26 2006

First column is central Delannoy numbers A001850. Second column is A050151.

			Triangle begins:
     1,
     3,     1,
    13,     6,     1,
    63,    33,     9,     1,
   321,   180,    62,    12,    1,
  1683,   985,   390,   100,   15,   1

Johann Cigler, Some elementary observations on Narayana polynomials and related topics, arXiv:1611.05252 [math.CO], 2016. See p. 19.
P. Peart and W.-J. Woan, Generating Functions via Hankel and Stieltjes Matrices, J. Integer Seqs., Vol. 3 (2000), #00.2.1.
P. Peart and W.-J. Woan, A divisibility property for a subgroup of Riordan matrices, Discrete Applied Mathematics, Vol. 98, Issue 3, Jan 2000, 255-263.
W.-J. Woan, Hankel Matrices and Lattice Paths, J. Integer Sequences, 4 (2001), #01.1.2.
Sheng-Liang Yang, Yan-Ni Dong, and Tian-Xiao He, Some matrix identities on colored Motzkin paths, Discrete Mathematics 340.12 (2017): 3081-3091.

Cf. A110171, A376467.

Table[Sum[Binomial[n,i]Binomial[n,n-k-i]2^i,{i,0,n-k}],{n,0,8},{k,0,8}]//MatrixForm

create_list(sum(binomial(n,i)*binomial(n,n-k-i)*2^i,i,0,n),n,0,8,k,0,n);

Number triangle T(n,k) = Sum_{j=0..n} C(n,j)*C(j,n-k-j)*2^(n-k-j)*3^(2*j-(n-k));
Riordan array (1/sqrt(1-6*x+x^2), (1-3*x-sqrt(1-6*x+x^2))/(4*x));
Column k has e.g.f. exp(3*x)*Bessel_I(k,2*sqrt(2)x)/(sqrt(2))^k.
a(n,k) = Sum_{i = 0..n} binomial(n,i)*binomial(n,n-k-i)*2^i, also a(n+1,k+1) = a(n,k) + 3*a(n,k+1) + 2*a(n,k+2). - Emanuele Munarini, Mar 16 2011
From Peter Bala, Jun 29 2015: (Start)
Matrix product A110171 * A007318.
Riordan array has the form ( x*h'(x)/h(x), h(x) ) with h(x) = ( 1 - 3*x - sqrt(1 - 6*x + x^2) )/(4*x) and so belongs to the hitting time subgroup H of the Riordan group (see Peart and Woan, Jan 2000, Example 5.2).
T(n,k) = [x^(n-k)] f(x)^n with f(x) = 1 + 3*x + 2*x^2. In general the (n,k)-th entry of the hitting time array ( x*h'(x)/h(x), h(x) ) has the form [x^(n-k)] f(x)^n, where f(x) = x/( series reversion of h(x) ). (End)

cp's OEIS Frontend

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

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