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
Examples
Triangle begins:
1,
3, 1,
13, 6, 1,
63, 33, 9, 1,
321, 180, 62, 12, 1,
1683, 985, 390, 100, 15, 1
Links
- 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.
Programs
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Mathematica
Table[Sum[Binomial[n,i]Binomial[n,n-k-i]2^i,{i,0,n-k}],{n,0,8},{k,0,8}]//MatrixForm -
Maxima
create_list(sum(binomial(n,i)*binomial(n,n-k-i)*2^i,i,0,n),n,0,8,k,0,n);
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)
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)
Comments