A059369 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.

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, 828, 2208, 5040, 1, 14, 96, 440, 1576, 4968, 14976, 40320, 1, 16, 126, 664, 2720, 9696, 33192, 115200, 362880, 1, 18, 160, 952, 4380, 17312, 64704, 247968, 996480

Offset: 1

Author

N. J. A. Sloane, Jan 28 2001

Keywords

• nonn
• tabl
• easy
• nice

Comments

Another version of triangle in A090238. - Philippe Deléham, Jun 14 2007

Examples

			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; ...
		

References

• L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 171, #34.

Crossrefs

Cf. A059370, A059371.

Programs

Mathematica

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 *)

Formula

G.f. for k-th diagonal: (Sum_{i >= 1} i!*t^i)^k = Sum_{n >= k} T(n, k)*t^n.

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

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Jan 31 2001