%I #18 Aug 05 2024 09:56:47

%S 1,5,6,19,24,30,67,86,110,140,227,294,380,490,630,751,978,1272,1652,

%T 2142,2772,2445,3196,4174,5446,7098,9240,12012,7869,10314,13510,17684,

%U 23130,30228,39468,51480

%N Triangle interpolating the swinging factorial (A056040) restricted to odd indices with its binomial inverse.

%C Triangle read by rows. For n >= 0, k >= 0 let

%C 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).

%H G. C. Greubel, <a href="/A163772/b163772.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 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.

%e 1

%e 5, 6

%e 19, 24, 30

%e 67, 86, 110, 140

%e 227, 294, 380, 490, 630

%e 751, 978, 1272, 1652, 2142, 2772

%e 2445, 3196, 4174, 5446, 7098, 9240, 12012

%p For the functions 'DiffTria' and 'swing' see A163770. Computes n rows of the triangle.

%p a := n -> DiffTria(k->swing(2*k+1),n,true);

%t 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 Row sums are A163775. Cf. A056040, A163650, A163771, A163772, A002426, A000984.

%K nonn,tabl

%O 0,2

%A _Peter Luschny_, Aug 05 2009