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

1,3

Comments

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

References

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

Links

Table of n, a(n) for n=1..54.

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

Example

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; ...

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 *)

Crossrefs

Cf. A059370, A059371.

Sequence in context: A062344 A208759 A033877 * A369518 A199530 A208765

Adjacent sequences: A059366 A059367 A059368 * A059370 A059371 A059372

Keyword

nonn,tabl,easy,nice

Author

N. J. A. Sloane, Jan 28 2001

Extensions

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

Status

approved