A101817 Triangle read by rows: T(n,h) = number of functions f:{1,2,...,n}->{1,2,...,n} such that |Image(f)|=h; h=1,2,...,n, n=1,2,3,... . Essentially A090657, but without zeros.

1, 2, 2, 3, 18, 6, 4, 84, 144, 24, 5, 300, 1500, 1200, 120, 6, 930, 10800, 23400, 10800, 720, 7, 2646, 63210, 294000, 352800, 105840, 5040, 8, 7112, 324576, 2857680, 7056000, 5362560, 1128960, 40320, 9, 18360, 1524600, 23496480, 105099120

Offset

1,2

Comments

Row sums = n^n. T(n,1) = n, T(n,n) = n!.

References

H. Picquet, Note #124, L'Intermédiaire des Mathématiciens, 1 (1894), pp. 125-127. - N. J. A. Sloane, Feb 28 2022

Links

Table of n, a(n) for n=1..41.

Formula

T(n, h) = C(n, h)*U(n, h), where U(n, h) is the array in A019538. Thus T(n, h) = C(n, h)*h!*S(n, h), where S(n, h) is a Stirling number of the second kind (given by A048993 with zeros removed).

T(2n,n) = A288312(n). - Alois P. Heinz, Jun 07 2017

Example

First rows:
1;
2,   2;
3,  18,   6;
4,  84, 144,  24;

Mathematica

Table[Table[StirlingS2[n, k] Binomial[n, k] k!, {k, 1, n}], {n, 1, 8}] // Grid

Crossrefs

Cf. A090657, A048993, A101818, A101819, A101821, A288312.

Sequence in context: A137909 A035796 A049009 * A058159 A058157 A348667

Adjacent sequences: A101814 A101815 A101816 * A101818 A101819 A101820

Keyword

nonn,tabl

Author

Clark Kimberling, Dec 17 2004

Status

approved