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

1, -1, 6, 1, -12, 30, -1, 18, -90, 140, 1, -24, 180, -560, 630, -1, 30, -300, 1400, -3150, 2772, 1, -36, 450, -2800, 9450, -16632, 12012, -1, 42, -630, 4900, -22050, 58212, -84084, 51480, 1, -48, 840, -7840, 44100, -155232, 336336, -411840, 218790

Offset: 0

Author

Peter Luschny, Aug 07 2009

Keywords

• sign
• tabl
• changed

Examples

			Triangle begins:
   1;
  -1,   6;
   1, -12,   30;
  -1,  18,  -90,   140;
   1, -24,  180,  -560,   630;
  -1,  30, -300,  1400, -3150,   2772;
   1, -36,  450, -2800,  9450, -16632, 12012;
		

Links

• Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
• Peter Luschny, Swinging Factorial.

Crossrefs

Row sums are the inverse binomial transform of the beta numbers (A163872).

Cf. A163649, A098473, A056040.

Programs

Maple

swing := proc(n) option remember; if n = 0 then 1 elif
irem(n, 2) = 1 then swing(n-1)*n else 4*swing(n-1)/n fi end:
a := proc(n, k) (-1)^(n-k)*binomial(n,k)*swing(2*k+1) end:
seq(print(seq(a(n,k),k=0..n)),n=0..8);

Mathematica

T[n_,k_] := ((-1)^(Mod[k,2]+n)*((2*k+1)!/(k!)^2)*Binomial[n,n-k]);
Flatten[Table[T[n,k],{n,0,8},{k,0,n}]] (* Detlef Meya, Oct 07 2023 *)

Formula

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).

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

T(n, k) = ((-1)^(k mod 2) + n)*((2*k + 1)!/(k!)^2)*binomial(n, n - k). - Detlef Meya, Oct 07 2023