A173333 Triangle read by rows: T(n, k) = n! / k!, 1 <= k <= n.

1, 2, 1, 6, 3, 1, 24, 12, 4, 1, 120, 60, 20, 5, 1, 720, 360, 120, 30, 6, 1, 5040, 2520, 840, 210, 42, 7, 1, 40320, 20160, 6720, 1680, 336, 56, 8, 1, 362880, 181440, 60480, 15120, 3024, 504, 72, 9, 1, 3628800, 1814400, 604800, 151200, 30240, 5040, 720, 90, 10, 1

Offset: 1

Author

Reinhard Zumkeller, Feb 19 2010

Keywords

• nonn
• tabl

Comments

From Wolfdieter Lang, Jun 27 2012: (Start)

T(n-1,k), k=1,...,n-1, gives the number of representative necklaces with n beads (C_N symmetry) of n+1-k distinct colors, say c[1],c[2],...,c[n-k+1], corresponding to the color signature determined by the partition k,1^(n-k) of n. The representative necklaces have k beads of color c[1]. E.g., n=4, k=2: partition 2,1,1, color signature (parts as exponents) c[1]c[1]c[2]c[3], 3=T(3,2) necklaces (write j for color c[j]): cyclic(1123), cyclic(1132) and cyclic(1213). See A212359 for the numbers for general partitions or color signatures. (End)

Examples

			Triangle starts:
n\k      1       2      3      4     5    6   7  8  9 10 ...
1        1
2        2       1
3        6       3      1
4       24      12      4      1
5      120      60     20      5     1
6      720     360    120     30     6    1
7     5040    2520    840    210    42    7   1
8    40320   20160   6720   1680   336   56   8  1
9   362880  181440  60480  15120  3024  504  72  9  1
10 3628800 1814400 604800 151200 30240 5040 720 90 10  1
... - _Wolfdieter Lang_, Jun 27 2012
		

Links

• Reinhard Zumkeller, Rows n = 1..150 of triangle, flattened
• Index entries for sequences related to factorial numbers.

Crossrefs

Row sums give A002627.

Central terms give A006963:

T(2*n-1,n) = A006963(n+1).

T(2*n,n) = A001813(n).

T(2*n,n+1) = A001761(n).

1 < k <= n: T(n,k) = T(n,k-1) / k.

1 <= k <= n: T(n+1,k) = A119741(n,n-k+1).

1 <= k <= n: T(n+1,k+1) = A162995(n,k).

T(n,1) = A000142(n).

T(n,2) = A001710(n) for n>1.

T(n,3) = A001715(n) for n>2.

T(n,4) = A001720(n) for n>3.

T(n,5) = A001725(n) for n>4.

T(n,6) = A001730(n) for n>5.

T(n,7) = A049388(n-7) for n>6.

T(n,8) = A049389(n-8) for n>7.

T(n,9) = A049398(n-9) for n>8.

T(n,10) = A051431(n) for n>9.

T(n,n-7) = A159083(n+1) for n>7.

T(n,n-6) = A053625(n+1) for n>6.

T(n,n-5) = A052787(n) for n>5.

T(n,n-4) = A052762(n) for n>4.

T(n,n-3) = A007531(n) for n>3.

T(n,n-2) = A002378(n-1) for n>2.

T(n,n-1) = A000027(n) for n>1.

T(n,n) = A000012(n).

Cf. A138533, A002627.

Programs

Haskell

a173333 n k = a173333_tabl !! (n-1) !! (k-1)
a173333_row n = a173333_tabl !! (n-1)
a173333_tabl = map fst $ iterate f ([1], 2)
   where f (row, i) = (map (* i) row ++ [1], i + 1)
-- Reinhard Zumkeller, Jul 04 2012

Mathematica

Table[n!/k!, {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Mar 01 2019 *)

Formula

E.g.f.: (exp(x*y) - 1)/(x*(1 - y)). - Olivier Gérard, Jul 07 2011

T(n,k) = A094587(n,k), 1 <= k <= n. - Reinhard Zumkeller, Jul 05 2012