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A269948 Triangle read by rows, Stirling set 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)+k^3*T(n-1, k), for n>=0 and 0<=k<=n.

%I A269948 #14 Jul 25 2019 08:34:51
%S A269948 1,0,1,0,1,1,0,1,9,1,0,1,73,36,1,0,1,585,1045,100,1,0,1,4681,28800,
%T A269948 7445,225,1,0,1,37449,782281,505280,35570,441,1,0,1,299593,21159036,
%U A269948 33120201,4951530,130826,784,1
%N A269948 Triangle read by rows, Stirling set 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)+k^3*T(n-1, k), for n>=0 and 0<=k<=n.
%C A269948 Also called 3rd central factorial numbers.
%F A269948 T(n,2) = (8^(n-1)-1)/7 for n>=1 (cf. A023001).
%F A269948 T(n,n-1) = (n*(n-1)/2)^2  for n>=1 (cf. A000537).
%F A269948 Row sums: A098437.
%e A269948 1,
%e A269948 0, 1,
%e A269948 0, 1, 1,
%e A269948 0, 1, 9,     1,
%e A269948 0, 1, 73,    36,     1,
%e A269948 0, 1, 585,   1045,   100,    1,
%e A269948 0, 1, 4681,  28800,  7445,   225,   1,
%e A269948 0, 1, 37449, 782281, 505280, 35570, 441, 1.
%p A269948 T := proc(n, k) option remember;
%p A269948     `if`(n=k, 1,
%p A269948     `if`(k<0 or k>n, 0,
%p A269948      T(n-1, k-1) + k^3*T(n-1, k))) end:
%p A269948 for n from 0 to 9 do seq(T(n,k), k=0..n) od;
%t A269948 T[n_, n_] = 1; T[n_, k_] /; 0 <= k <= n := T[n, k] = T[n - 1, k - 1] + k^3*T[n - 1, k]; T[_, _] = 0;
%t A269948 Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jul 25 2019 *)
%Y A269948 Variant: A098436.
%Y A269948 Cf. A007318 (order 0), A048993 (order 1), A269945 (order 2).
%Y A269948 Cf. A000537, A023001, A098437.
%K A269948 nonn,tabl
%O A269948 0,9
%A A269948 _Peter Luschny_, Mar 22 2016