A269947 - OEIS

%I #12 Jul 12 2019 15:42:19

%S 1,0,1,0,1,1,0,8,9,1,0,216,251,36,1,0,13824,16280,2555,100,1,0,

%T 1728000,2048824,335655,15055,225,1,0,373248000,444273984,74550304,

%U 3587535,63655,441,1,0,128024064000,152759224512,26015028256,1305074809,25421200,214918,784,1

%N Triangle read by rows, Stirling cycle numbers of order 3, T(n,n) = 1, T(n,k) = 0 if k<0 or k>n, otherwise T(n,k) = T(n-1,k-1)+(n-1)^3*T(n-1,k), for n>=0 and 0<=k<=n.

%H Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/P-Transform">The P-transform</a>.

%F T(n,1) = ((n-1)!)^3 for n>=1 (cf. A000442).

%F T(n,n-1) = (n*(n-1)/2)^2 for n>=1 (cf. A000537).

%F Row sums: Product_{k=1..n} ((k-1)^3+1) for n>=0 (cf. A255433).

%e Triangle starts:

%e 1,

%e 0, 1,

%e 0, 1,       1,

%e 0, 8,       9,       1,

%e 0, 216,     251,     36,     1,

%e 0, 13824,   16280,   2555,   100,   1,

%e 0, 1728000, 2048824, 335655, 15055, 225, 1.

%p T := proc(n, k) option remember;

%p     `if`(n=k, 1,

%p     `if`(k<0 or k>n, 0,

%p      T(n-1, k-1) + (n-1)^3*T(n-1, k))) end:

%p for n from 0 to 6 do seq(T(n,k), k=0..n) od;

%t T[n_, k_] := T[n, k] = Which[n == k, 1, k < 0 || k > n, 0, True, T[n - 1, k - 1] + (n - 1)^3 T[n - 1, k]];

%t Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jul 12 2019 *)

%Y Variant: A249677.

%Y Cf. A007318 (order 0), A132393 (order 1), A269944 (order 2).

%Y Cf. A000442, A000537, A255433.

%K nonn,tabl

%O 0,8

%A _Peter Luschny_, Mar 22 2016