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%I #10 May 08 2020 17:51:34
%S 1,0,1,0,2,8,0,6,48,162,0,6,96,486,1536,0,30,960,7290,30720,93750,0,
%T 20,1280,14580,81920,312500,933120,0,140,17920,306180,2293760,
%U 10937500,39191040,115296020,0,70,17920,459270,4587520,27343750,117573120,403536070,1174405120
%N Triangle read by rows, T(n,k) = k^n*A056040(n), n>=0, 0<=k<=n.
%H Peter Luschny, <a href="/A180000/a180000.pdf">Die schwingende Fakultät und Orbitalsysteme</a>, August 2011.
%F 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.
%e 1
%e 0, 1
%e 0, 2, 8
%e 0, 6, 48, 162
%e 0, 6, 96, 486, 1536
%e 0, 30, 960, 7290, 30720, 93750
%e 0, 20, 1280, 14580, 81920, 312500, 933120
%p 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):
%p # Formula:
%p omega := proc(x) BesselI(0,2*m*x)+(2*m*x+1)*BesselI(1,2*m*x) end:
%p f := n -> `if`(irem(n,2)=1,(n+1)/2,1/(n+1)): A195009 := proc(n,k)
%p limit(f(n)*(D@@n)(omega)(x),x=0); subs(m=k,%) end;
%t 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 *)
%K nonn,tabl
%O 0,5
%A _Peter Luschny_, Sep 07 2011
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