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A163841 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).

%I A163841 #25 Aug 22 2025 04:25:21
%S A163841 1,3,2,11,8,6,45,34,26,20,195,150,116,90,70,873,678,528,412,322,252,
%T A163841 3989,3116,2438,1910,1498,1176,924,18483,14494,11378,8940,7030,5532,
%U A163841 4356,3432,86515,68032,53538
%N A163841 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 A163841 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 A163841 G. C. Greubel, <a href="/A163841/b163841.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H A163841 Peter Luschny, <a href="/A180000/a180000.pdf">Die schwingende Fakultät und Orbitalsysteme</a>, August 2011.
%H A163841 Peter Luschny, <a href="http://www.luschny.de/math/swing/SwingingFactorial.html">Swinging Factorial</a>.
%H A163841 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 A163841 Triangle begins
%e A163841      1;
%e A163841      3,    2;
%e A163841     11,    8,    6;
%e A163841     45,   34,   26,   20;
%e A163841    195,  150,  116,   90,   70;
%e A163841    873,  678,  528,  412,  322,  252;
%e A163841   3989, 3116, 2438, 1910, 1498, 1176,  924;
%p A163841 Computes n rows of the triangle. For the functions 'SumTria' and 'swing' see A163840.
%p A163841 a := n -> SumTria(k->swing(2*k),n,true);
%t A163841 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 A163841 Row sums are A163844. Cf. A056040, A163650, A163841, A163842, A163840, A026375, A002426, A000984.
%K A163841 nonn,tabl,changed
%O A163841 0,2
%A A163841 _Peter Luschny_, Aug 06 2009