OFFSET
0,3
COMMENTS
Triangle read by rows.
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).
LINKS
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Factorial.
FORMULA
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
EXAMPLE
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;
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
From Detlef Meya, Oct 07 2023: (Start)
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}]] (*End*)
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Peter Luschny, Aug 07 2009
STATUS
approved