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A163842 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). Triangle read by rows, for n >= 0, k >= 0.

%I A163842 #23 Aug 22 2025 04:32:13
%S A163842 1,7,6,43,36,30,249,206,170,140,1395,1146,940,770,630,7653,6258,5112,
%T A163842 4172,3402,2772,41381,33728,27470,22358,18186,14784,12012,221399,
%U A163842 180018,146290,118820,96462,78276,63492,51480
%N A163842 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). Triangle read by rows, for n >= 0, k >= 0.
%H A163842 G. C. Greubel, <a href="/A163842/b163842.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H A163842 Peter Luschny, <a href="/A180000/a180000.pdf">Die schwingende Fakultät und Orbitalsysteme</a>, August 2011.
%H A163842 Peter Luschny, <a href="http://www.luschny.de/math/swing/SwingingFactorial.html">Swinging Factorial</a>.
%F A163842 T(n,k) = Sum_{i=k..n} binomial(n-k,n-i)*(2i+1)$ where i$ denotes the swinging factorial of i (A056040).
%e A163842 Triangle begins:
%e A163842       1;
%e A163842       7,     6;
%e A163842      43,    36,    30;
%e A163842     249,   206,   170,   140;
%e A163842    1395,  1146,   940,   770,   630;
%e A163842    7653,  6258,  5112,  4172,  3402,  2772;
%e A163842   41381, 33728, 27470, 22358, 18186, 14784, 12012;
%p A163842 # Computes n rows of the triangle. For the functions 'SumTria' and 'swing' see A163840.
%p A163842 a := n -> SumTria(k->swing(2*k+1),n,true);
%t A163842 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 A163842 Row sums are A163845. Cf. A056040, A163650, A163841, A163842, A163840, A002426, A000984.
%K A163842 nonn,tabl,changed
%O A163842 0,2
%A A163842 _Peter Luschny_, Aug 06 2009