A225723 - OEIS

%I #11 Dec 17 2021 18:57:34

%S 1,2,3,12,9,17,108,72,68,142,1280,810,680,710,1569,18750,11520,9180,

%T 8520,9414,21576,326592,196875,152320,134190,131796,151032,355081,

%U 6588344,3919104,2975000,2544640,2372328,2416512,2840648,6805296

%N Triangular array read by rows: T(n,k) is the number of size k components in the digraph representation of all functions f:{1,2,...,n}->{1,2,...,n}; n>=1, 1<=k<=n.

%C T(n,1) = n*(n-1)^(n-1) = A055897(n).

%C Row sums = A190314.

%C T(n,n) = A001865(n).

%C Sum_{k=1..n} T(n,k)*k = n^(n+1).

%H Alois P. Heinz, <a href="/A225723/b225723.txt">Rows n = 1..100, flattened</a>

%F E.g.f.: log(1/(1 - A(x*y)))/(1 - A(x)) where A(x) is the e.g.f. for A000169.

%F T(n,k) = C(n,k)*A001865(k)*A000312(n-k). - _Alois P. Heinz_, May 13 2013

%e Triangle T(n,k) begins:

%e        1;

%e        2,      3;

%e       12,      9,     17;

%e      108,     72,     68,    142;

%e     1280,    810,    680,    710,   1569;

%e    18750,  11520,   9180,   8520,   9414,  21576;

%e   326592, 196875, 152320, 134190, 131796, 151032, 355081;

%e   ...

%p b:= n-> n!*add(n^(n-k-1)/(n-k)!, k=1..n):

%p T:= (n, k)-> binomial(n,k)*b(k)*(n-k)^(n-k):

%p seq(seq(T(n, k), k=1..n), n=1..10);  # _Alois P. Heinz_, May 13 2013

%t nn = 8; tx = Sum[n^(n - 1) x^n/n!, {n, 1, nn}]; txy =

%t Sum[n^(n - 1) (x y)^n/n!, {n, 1, nn}];

%t Map[Select[#, # > 0 &] &,

%t   Drop[Range[0, nn]! CoefficientList[

%t      Series[Log[1/(1 - txy)]/(1 - tx), {x, 0, nn}], {x, y}],

%t    1]] // Grid

%Y Cf. A225213.

%K nonn,tabl

%O 1,2

%A _Geoffrey Critzer_, May 13 2013