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%I #16 Jan 22 2020 14:48:24
%S 1,1,2,1,4,6,1,6,16,24,1,8,30,72,120,1,10,48,152,372,720,1,12,70,272,
%T 828,2208,5040,1,14,96,440,1576,4968,14976,40320,1,16,126,664,2720,
%U 9696,33192,115200,362880,1,18,160,952,4380,17312,64704,247968,996480
%N Triangle of numbers T(n,k) = T(n-1,k-1) + ((n+k-1)/k)*T(n-1,k), n >= 1, 1 <= k <= n, with T(n,1) = n!, T(n,n) = 1; read from right to left.
%C Another version of triangle in A090238. - _Philippe Deléham_, Jun 14 2007
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 171, #34.
%F G.f. for k-th diagonal: (Sum_{i >= 1} i!*t^i)^k = Sum_{n >= k} T(n, k)*t^n.
%F T(n,k) = n! if k=1, 1 if k=n, Sum_{m=0..n-k} (m+1)!*T(n-m-1,k-1) otherwise. - _Vladimir Kruchinin_, Aug 18 2010
%e When read from left to right the rows {T(n,k), 1 <= k <= n} for n=1,2,3,... are 1; 2,1; 6,4,1; 24,16,6,1; ...
%t nmax = 10; t[n_, k_] := Sum[(m+1)!*t[n-m-1, k-1], {m, 0, n-k}]; t[n_, 1] = n!; t[n_, n_] = 1; Flatten[ Table[ t[n, k], {n, 1, nmax}, {k, n, 1, -1}]] (* _Jean-François Alcover_, Nov 14 2011 *)
%Y Cf. A059370, A059371.
%K nonn,tabl,easy,nice
%O 1,3
%A _N. J. A. Sloane_, Jan 28 2001
%E More terms from Larry Reeves (larryr(AT)acm.org), Jan 31 2001