cp's OEIS Frontend

A163945 Triangle interpolating between (-1)^n (A033999) and the swinging factorial function (A056040) restricted to odd indices (2n+1)$ (A002457), read by rows.

%I A163945 #30 Aug 22 2025 04:30:50
%S A163945 1,-1,6,1,-12,30,-1,18,-90,140,1,-24,180,-560,630,-1,30,-300,1400,
%T A163945 -3150,2772,1,-36,450,-2800,9450,-16632,12012,-1,42,-630,4900,-22050,
%U A163945 58212,-84084,51480,1,-48,840,-7840,44100,-155232,336336,-411840,218790
%N A163945 Triangle interpolating between (-1)^n (A033999) and the swinging factorial function (A056040) restricted to odd indices (2n+1)$ (A002457), read by rows.
%H A163945 Peter Luschny, <a href="/A180000/a180000.pdf">Die schwingende Fakultät und Orbitalsysteme</a>, August 2011.
%H A163945 Peter Luschny, <a href="http://www.luschny.de/math/swing/SwingingFactorial.html">Swinging Factorial</a>.
%F A163945 For n >= 0, k >= 0, T(n, k) = (-1)^(n-k) binomial(n,k) (2*k+1)$ where i$ denotes the swinging factorial of i (A056040).
%F A163945 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 A163945 T(n, k) = ((-1)^(k mod 2) + n)*((2*k + 1)!/(k!)^2)*binomial(n, n - k). - _Detlef Meya_, Oct 07 2023
%e A163945 Triangle begins:
%e A163945    1;
%e A163945   -1,   6;
%e A163945    1, -12,   30;
%e A163945   -1,  18,  -90,   140;
%e A163945    1, -24,  180,  -560,   630;
%e A163945   -1,  30, -300,  1400, -3150,   2772;
%e A163945    1, -36,  450, -2800,  9450, -16632, 12012;
%p A163945 swing := proc(n) option remember; if n = 0 then 1 elif
%p A163945 irem(n, 2) = 1 then swing(n-1)*n else 4*swing(n-1)/n fi end:
%p A163945 a := proc(n, k) (-1)^(n-k)*binomial(n,k)*swing(2*k+1) end:
%p A163945 seq(print(seq(a(n,k),k=0..n)),n=0..8);
%t A163945 T[n_,k_] := ((-1)^(Mod[k,2]+n)*((2*k+1)!/(k!)^2)*Binomial[n,n-k]);
%t A163945 Flatten[Table[T[n,k],{n,0,8},{k,0,n}]] (* _Detlef Meya_, Oct 07 2023 *)
%Y A163945 Row sums are the inverse binomial transform of the beta numbers (A163872).
%Y A163945 Cf. A163649, A098473, A056040.
%K A163945 sign,tabl,changed
%O A163945 0,3
%A A163945 _Peter Luschny_, Aug 07 2009