%I #21 May 08 2020 17:41:21
%S 1,3,2,11,8,6,45,34,26,20,195,150,116,90,70,873,678,528,412,322,252,
%T 3989,3116,2438,1910,1498,1176,924,18483,14494,11378,8940,7030,5532,
%U 4356,3432,86515,68032,53538
%N Triangle interpolating the swinging factorial (A056040) restricted to even indices with its binomial transform. Same as interpolating bilateral Schroeder paths (A026375) with the central binomial coefficients (A000984).
%C For n >= 0, k >= 0 let T(n,k) = sum{i=k..n} binomial(n-k,n-i)*(2i)$ where i$ denotes the swinging factorial of i (A056040). Triangle read by rows.
%H G. C. Greubel, <a href="/A163841/b163841.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H Peter Luschny, <a href="/A180000/a180000.pdf">Die schwingende Fakultät und Orbitalsysteme</a>, August 2011.
%H Peter Luschny, <a href="http://www.luschny.de/math/swing/SwingingFactorial.html"> Swinging Factorial</a>.
%H Tony D. Noe, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Noe/noe35.html">On the Divisibility of Generalized Central Trinomial Coefficients</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.
%e Triangle begins
%e 1;
%e 3, 2;
%e 11, 8, 6;
%e 45, 34, 26, 20;
%e 195, 150, 116, 90, 70;
%e 873, 678, 528, 412, 322, 252;
%e 3989, 3116, 2438, 1910, 1498, 1176, 924;
%p Computes n rows of the triangle. For the functions 'SumTria' and 'swing' see A163840.
%p a := n -> SumTria(k->swing(2*k),n,true);
%t sf[n_] := n!/Quotient[n, 2]!^2; t[n_, k_] := Sum[Binomial[n - k, n - i]*sf[2*i], {i, k, n}]; Table[t[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jun 28 2013 *)
%Y Row sums are A163844. Cf. A056040, A163650, A163841, A163842, A163840, A026375, A002426, A000984.
%K nonn,tabl
%O 0,2
%A _Peter Luschny_, Aug 06 2009