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%I #19 Oct 07 2023 06:26:05
%S 1,-1,6,1,-12,30,-1,18,-90,140,1,-24,180,-560,630,-1,30,-300,1400,
%T -3150,2772,1,-36,450,-2800,9450,-16632,12012,-1,42,-630,4900,-22050,
%U 58212,-84084,51480,1,-48,840,-7840,44100,-155232,336336,-411840,218790
%N Triangle interpolating between (-1)^n (A033999) and the swinging factorial function (A056040) restricted to odd indices (2n+1)$ (A002457).
%C Triangle read by rows.
%C For n >= 0, k >= 0 let T(n, k) = (-1)^(n-k) binomial(n,k) (2*k+1)$ where i$ denotes the swinging factorial of i (A056040).
%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>.
%F Conjectural g.f.: sqrt(1 + t)/(1 + (1 - 4*x)*t)^(3/2) = 1 + (-1 + 6*x)*t + (1 - 12*x + 30*x^2)*t^2 + .... - _Peter Bala_, Nov 10 2013
%F T(n, k) = ((-1)^(k mod 2) + n)*((2*k + 1)!/(k!)^2)*binomial(n, n - k). - _Detlef Meya_, Oct 07 2023
%e 1;
%e -1, 6;
%e 1, -12, 30;
%e -1, 18, -90, 140;
%e 1, -24, 180, -560, 630;
%e -1, 30, -300, 1400, -3150, 2772;
%e 1, -36, 450, -2800, 9450, -16632, 12012;
%p swing := proc(n) option remember; if n = 0 then 1 elif
%p irem(n, 2) = 1 then swing(n-1)*n else 4*swing(n-1)/n fi end:
%p a := proc(n, k) (-1)^(n-k)*binomial(n,k)*swing(2*k+1) end:
%p seq(print(seq(a(n,k),k=0..n)),n=0..8);
%t From _Detlef Meya_, Oct 07 2023: (Start)
%t T[n_,k_] := ((-1)^(Mod[k,2]+n)*((2*k+1)!/(k!)^2)*Binomial[n,n-k]);
%t Flatten[Table[T[n,k],{n,0,8},{k,0,n}]] (*End*)
%Y Row sums are the inverse binomial transform of the beta numbers (A163872).
%Y Cf. A163649, A098473, A056040.
%K sign,tabl
%O 0,3
%A _Peter Luschny_, Aug 07 2009
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