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

%I A118384 #43 Jan 24 2025 08:50:39
%S A118384 1,3,1,13,6,1,63,33,9,1,321,180,62,12,1,1683,985,390,100,15,1,8989,
%T A118384 5418,2355,720,147,18,1,48639,29953,13923,4809,1197,203,21,1,265729,
%U A118384 166344,81340,30744,8806,1848,268,24,1,1462563,927441,471852,191184,60858
%N A118384 Gaussian column reduction of Hankel matrix for central Delannoy numbers.
%C A118384 First column is central Delannoy numbers A001850. Second column is A050151.
%H A118384 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 A118384 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 A118384 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 A118384 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 A118384 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 A118384 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));
%F A118384 Riordan array (1/sqrt(1-6*x+x^2), (1-3*x-sqrt(1-6*x+x^2))/(4*x));
%F A118384 Column k has e.g.f. exp(3*x)*Bessel_I(k,2*sqrt(2)x)/(sqrt(2))^k.
%F A118384 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
%F A118384 From _Peter Bala_, Jun 29 2015: (Start)
%F A118384 Matrix product A110171 * A007318.
%F A118384 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 A118384 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 A118384 Triangle begins:
%e A118384      1,
%e A118384      3,     1,
%e A118384     13,     6,     1,
%e A118384     63,    33,     9,     1,
%e A118384    321,   180,    62,    12,    1,
%e A118384   1683,   985,   390,   100,   15,   1
%t A118384 Table[Sum[Binomial[n,i]Binomial[n,n-k-i]2^i,{i,0,n-k}],{n,0,8},{k,0,8}]//MatrixForm
%o A118384 (Maxima) create_list(sum(binomial(n,i)*binomial(n,n-k-i)*2^i,i,0,n),n,0,8,k,0,n);
%Y A118384 Cf. A110171, A376467.
%K A118384 easy,nonn,tabl
%O A118384 0,2
%A A118384 _Paul Barry_, Apr 26 2006