A101817 - OEIS

%I #15 Mar 03 2022 12:43:49

%S 1,2,2,3,18,6,4,84,144,24,5,300,1500,1200,120,6,930,10800,23400,10800,

%T 720,7,2646,63210,294000,352800,105840,5040,8,7112,324576,2857680,

%U 7056000,5362560,1128960,40320,9,18360,1524600,23496480,105099120

%N 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.

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

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

%F 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).

%F T(2n,n) = A288312(n). - _Alois P. Heinz_, Jun 07 2017

%e First rows:

%e 1;

%e 2,   2;

%e 3,  18,   6;

%e 4,  84, 144,  24;

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

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

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Dec 17 2004