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%I #14 May 08 2020 17:39:21
%S 1,7,6,43,36,30,249,206,170,140,1395,1146,940,770,630,7653,6258,5112,
%T 4172,3402,2772,41381,33728,27470,22358,18186,14784,12012,221399,
%U 180018,146290,118820,96462,78276,63492
%N Triangle interpolating the swinging factorial (A056040) restricted to odd indices with its binomial transform. Same as interpolating the beta numbers 1/beta(n,n) (A002457) with (A163869).
%C Triangle read by rows. For n >= 0, k >= 0 let
%C T(n,k) = sum{i=k..n} binomial(n-k,n-i)*(2i+1)$
%C where i$ denotes the swinging factorial of i (A056040).
%H G. C. Greubel, <a href="/A163842/b163842.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>.
%e 1
%e 7, 6
%e 43, 36, 30
%e 249, 206, 170, 140
%e 1395, 1146, 940, 770, 630
%e 7653, 6258, 5112, 4172, 3402, 2772
%e 41381, 33728, 27470, 22358, 18186, 14784, 12012
%p Computes n rows of the triangle. For the functions 'SumTria' and 'swing' see A163840.
%p a := n -> SumTria(k->swing(2*k+1),n,true);
%t sf[n_] := n!/Quotient[n, 2]!^2; t[n_, k_] := Sum[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 A163845. Cf. A056040, A163650, A163841, A163842, A163840, A002426, A000984.
%K nonn,tabl
%O 0,2
%A _Peter Luschny_, Aug 06 2009
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