A195009 Triangle read by rows, T(n,k) = k^n*A056040(n), n>=0, 0<=k<=n.

1, 0, 1, 0, 2, 8, 0, 6, 48, 162, 0, 6, 96, 486, 1536, 0, 30, 960, 7290, 30720, 93750, 0, 20, 1280, 14580, 81920, 312500, 933120, 0, 140, 17920, 306180, 2293760, 10937500, 39191040, 115296020, 0, 70, 17920, 459270, 4587520, 27343750, 117573120, 403536070, 1174405120

Offset: 0

Author

Peter Luschny, Sep 07 2011

Keywords

• nonn
• tabl

Examples

			                     1
                    0, 1
                  0, 2, 8
               0, 6, 48, 162
            0, 6, 96, 486, 1536
       0, 30, 960, 7290, 30720, 93750
0, 20, 1280, 14580, 81920, 312500, 933120
		

Links

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

Programs

Maple

swing := n -> n!/iquo(n,2)!^2: pow := (n,k) -> if k=0 and n=0 then 1 else n^k fi: A195009 := (n,k) -> pow(k,n)*swing(n):
# Formula:
omega := proc(x) BesselI(0,2*m*x)+(2*m*x+1)*BesselI(1,2*m*x) end:
f := n -> `if`(irem(n,2)=1,(n+1)/2,1/(n+1)): A195009 := proc(n,k)
limit(f(n)*(D@@n)(omega)(x),x=0); subs(m=k,%) end;

Mathematica

sf[n_] := With[{f = Floor[n/2]}, Pochhammer[f+1, n-f]/f!]; t[0, 0] = 1; t[n_, k_] := k^n*sf[n]; Table[t[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 29 2013 *)

Formula

T(n,k) = f(n)*lim(x=0, (d^n/dx)(BesselI(0,2*k*x)+(2*k*x+1) *BesselI(1,2*k*x) where f(n) = (n+1)/2 if n is odd, 1/(n+1) otherwise.