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A225723 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.

%I A225723 #11 Dec 17 2021 18:57:34
%S A225723 1,2,3,12,9,17,108,72,68,142,1280,810,680,710,1569,18750,11520,9180,
%T A225723 8520,9414,21576,326592,196875,152320,134190,131796,151032,355081,
%U A225723 6588344,3919104,2975000,2544640,2372328,2416512,2840648,6805296
%N A225723 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 A225723 T(n,1) = n*(n-1)^(n-1) = A055897(n).
%C A225723 Row sums = A190314.
%C A225723 T(n,n) = A001865(n).
%C A225723 Sum_{k=1..n} T(n,k)*k = n^(n+1).
%H A225723 Alois P. Heinz, <a href="/A225723/b225723.txt">Rows n = 1..100, flattened</a>
%F A225723 E.g.f.: log(1/(1 - A(x*y)))/(1 - A(x)) where A(x) is the e.g.f. for A000169.
%F A225723 T(n,k) = C(n,k)*A001865(k)*A000312(n-k). - _Alois P. Heinz_, May 13 2013
%e A225723 Triangle T(n,k) begins:
%e A225723        1;
%e A225723        2,      3;
%e A225723       12,      9,     17;
%e A225723      108,     72,     68,    142;
%e A225723     1280,    810,    680,    710,   1569;
%e A225723    18750,  11520,   9180,   8520,   9414,  21576;
%e A225723   326592, 196875, 152320, 134190, 131796, 151032, 355081;
%e A225723   ...
%p A225723 b:= n-> n!*add(n^(n-k-1)/(n-k)!, k=1..n):
%p A225723 T:= (n, k)-> binomial(n,k)*b(k)*(n-k)^(n-k):
%p A225723 seq(seq(T(n, k), k=1..n), n=1..10);  # _Alois P. Heinz_, May 13 2013
%t A225723 nn = 8; tx = Sum[n^(n - 1) x^n/n!, {n, 1, nn}]; txy =
%t A225723 Sum[n^(n - 1) (x y)^n/n!, {n, 1, nn}];
%t A225723 Map[Select[#, # > 0 &] &,
%t A225723   Drop[Range[0, nn]! CoefficientList[
%t A225723      Series[Log[1/(1 - txy)]/(1 - tx), {x, 0, nn}], {x, y}],
%t A225723    1]] // Grid
%Y A225723 Cf. A225213.
%K A225723 nonn,tabl
%O A225723 1,2
%A A225723 _Geoffrey Critzer_, May 13 2013