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

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

0,2

Comments

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

Links

Table of n, a(n) for n=0..49.

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.

Formula

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)

Example

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

Mathematica

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

Prog

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

Crossrefs

Cf. A110171, A376467.

Sequence in context: A266577 A143411 A096773 * A341725 A258239 A376863

Adjacent sequences: A118381 A118382 A118383 * A118385 A118386 A118387

Keyword

easy,nonn,tabl

Author

Paul Barry, Apr 26 2006

Status

approved