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%I #35 May 04 2019 04:42:02
%S 1,3,1,13,6,1,63,33,9,1,321,180,62,12,1,1683,985,390,100,15,1,8989,
%T 5418,2355,720,147,18,1,48639,29953,13923,4809,1197,203,21,1,265729,
%U 166344,81340,30744,8806,1848,268,24,1,1462563,927441,471852,191184,60858
%N Gaussian column reduction of Hankel matrix for central Delannoy numbers.
%C First column is central Delannoy numbers A001850. Second column is A050151.
%H Johann Cigler, <a href="https://arxiv.org/abs/1611.05252">Some elementary observations on Narayana polynomials and related topics</a>, arXiv:1611.05252 [math.CO], 2016. See p. 19.
%H P. Peart and W.-J. Woan, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/PEART/peart1.html">Generating Functions via Hankel and Stieltjes Matrices</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.1.
%H P. Peart and W.-J. Woan, <a href="http://dx.doi.org/10.1016/S0166-218X(99)00166-3">A divisibility property for a subgroup of Riordan matrices</a>, Discrete Applied Mathematics, Vol. 98, Issue 3, Jan 2000, 255-263.
%H W.-J. Woan, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL4/WOAN/hankel2.html">Hankel Matrices and Lattice Paths</a>, J. Integer Sequences, 4 (2001), #01.1.2.
%H Sheng-Liang Yang, Yan-Ni Dong, and Tian-Xiao He, <a href="https://doi.org/10.1016/j.disc.2017.07.006">Some matrix identities on colored Motzkin paths</a>, Discrete Mathematics 340.12 (2017): 3081-3091.
%F 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.
%F 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
%F From _Peter Bala_, Jun 29 2015: (Start)
%F Matrix product A110171 * A007318.
%F 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).
%F 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)
%e Triangle begins
%e 1,
%e 3, 1,
%e 13, 6, 1,
%e 63, 33, 9, 1,
%e 321, 180, 62, 12, 1,
%e 1683, 985, 390, 100, 15, 1
%t Table[Sum[Binomial[n,i]Binomial[n,n-k-i]2^i,{i,0,n-k}],{n,0,8},{k,0,8}]//MatrixForm
%o (Maxima) create_list(sum(binomial(n,i)*binomial(n,n-k-i)*2^i,i,0,n),n,0,8,k,0,n);
%Y Cf. A110171.
%K easy,nonn,tabl
%O 0,2
%A _Paul Barry_, Apr 26 2006
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