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A118384 Gaussian column reduction of Hankel matrix for central Delannoy numbers. 5
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 (list; table; graph; refs; listen; history; text; internal format)
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^(2j-(n-k))}; Riordan array (1/sqrt(1-6x+x^2), (1-3x-sqrt(1-6x+x^2))/(4x)); Column k has e.g.f. exp(3x)Bessel_I(k,2*sqrt(2)x)/(sqrt(2))^k.

a(n) = sum(binomial(n,i)*binomial(n,n-k-i)*2^i,i=0..n), 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.

Sequence in context: A266577 A143411 A096773 * A258239 A133176 A089435

Adjacent sequences:  A118381 A118382 A118383 * A118385 A118386 A118387

KEYWORD

easy,nonn,tabl

AUTHOR

Paul Barry, Apr 26 2006

STATUS

approved

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Last modified July 14 18:26 EDT 2020. Contains 335729 sequences. (Running on oeis4.)