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A163772 Triangle interpolating the swinging factorial (A056040) restricted to odd indices with its binomial inverse. Triangle read by rows. For n >= 0, k >= 0.

%I A163772 #28 Aug 22 2025 04:32:29
%S A163772 1,5,6,19,24,30,67,86,110,140,227,294,380,490,630,751,978,1272,1652,
%T A163772 2142,2772,2445,3196,4174,5446,7098,9240,12012,7869,10314,13510,17684,
%U A163772 23130,30228,39468,51480
%N A163772 Triangle interpolating the swinging factorial (A056040) restricted to odd indices with its binomial inverse. Triangle read by rows. For n >= 0, k >= 0.
%H A163772 G. C. Greubel, <a href="/A163772/b163772.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H A163772 Peter Luschny, <a href="/A180000/a180000.pdf">Die schwingende Fakultät und Orbitalsysteme</a>, August 2011.
%H A163772 Peter Luschny, <a href="http://www.luschny.de/math/swing/SwingingFactorial.html">Swinging Factorial</a>.
%H A163772 M. Z. Spivey and L. L. Steil, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Spivey/spivey7.html">The k-Binomial Transforms and the Hankel Transform</a>, J. Integ. Seqs. Vol. 9 (2006), #06.1.1.
%F A163772 T(n,k) = Sum_{i=k..n} (-1)^(n-i)*binomial(n-k,n-i)*(2i+1)$  where i$ denotes the swinging factorial of i (A056040).
%e A163772 Triangle begins:
%e A163772      1;
%e A163772      5,    6;
%e A163772     19,   24,   30;
%e A163772     67,   86,  110,  140;
%e A163772    227,  294,  380,  490,  630;
%e A163772    751,  978, 1272, 1652, 2142, 2772;
%e A163772   2445, 3196, 4174, 5446, 7098, 9240, 12012;
%p A163772 For the functions 'DiffTria' and 'swing' see A163770. Computes n rows of the triangle.
%p A163772 a := n -> DiffTria(k->swing(2*k+1),n,true);
%t A163772 sf[n_] := n!/Quotient[n, 2]!^2; t[n_, k_] := Sum[ (-1)^(n-i)*Binomial[n-k, n-i]*sf[2*i+1], {i, k, n}]; Table[t[n, k], {n, 0, 7}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jun 28 2013 *)
%Y A163772 Row sums are A163775. Cf. A056040, A163650, A163771, A163772, A002426, A000984.
%K A163772 nonn,tabl,changed
%O A163772 0,2
%A A163772 _Peter Luschny_, Aug 05 2009