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A269947 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.

%I A269947 #12 Jul 12 2019 15:42:19
%S A269947 1,0,1,0,1,1,0,8,9,1,0,216,251,36,1,0,13824,16280,2555,100,1,0,
%T A269947 1728000,2048824,335655,15055,225,1,0,373248000,444273984,74550304,
%U A269947 3587535,63655,441,1,0,128024064000,152759224512,26015028256,1305074809,25421200,214918,784,1
%N A269947 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 A269947 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/P-Transform">The P-transform</a>.
%F A269947 T(n,1) = ((n-1)!)^3 for n>=1 (cf. A000442).
%F A269947 T(n,n-1) = (n*(n-1)/2)^2 for n>=1 (cf. A000537).
%F A269947 Row sums: Product_{k=1..n} ((k-1)^3+1) for n>=0 (cf. A255433).
%e A269947 Triangle starts:
%e A269947 1,
%e A269947 0, 1,
%e A269947 0, 1,       1,
%e A269947 0, 8,       9,       1,
%e A269947 0, 216,     251,     36,     1,
%e A269947 0, 13824,   16280,   2555,   100,   1,
%e A269947 0, 1728000, 2048824, 335655, 15055, 225, 1.
%p A269947 T := proc(n, k) option remember;
%p A269947     `if`(n=k, 1,
%p A269947     `if`(k<0 or k>n, 0,
%p A269947      T(n-1, k-1) + (n-1)^3*T(n-1, k))) end:
%p A269947 for n from 0 to 6 do seq(T(n,k), k=0..n) od;
%t A269947 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 A269947 Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jul 12 2019 *)
%Y A269947 Variant: A249677.
%Y A269947 Cf. A007318 (order 0), A132393 (order 1), A269944 (order 2).
%Y A269947 Cf. A000442, A000537, A255433.
%K A269947 nonn,tabl
%O A269947 0,8
%A A269947 _Peter Luschny_, Mar 22 2016