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A201685 Triangular array read by rows. T(n,k) is the number of connected endofunctions on {1,2,...,n} that have exactly k nodes in the unique cycle of its digraph representation.

%I A201685 #34 Sep 08 2022 08:46:01
%S A201685 1,2,1,9,6,2,64,48,24,6,625,500,300,120,24,7776,6480,4320,2160,720,
%T A201685 120,117649,100842,72030,41160,17640,5040,720,2097152,1835008,1376256,
%U A201685 860160,430080,161280,40320,5040,43046721,38263752,29760696,19840464,11022480,4898880,1632960,362880,40320
%N A201685 Triangular array read by rows. T(n,k) is the number of connected endofunctions on {1,2,...,n} that have exactly k nodes in the unique cycle of its digraph representation.
%C A201685 Column k=1: A000169,
%C A201685 Column k=2: A053506,
%C A201685 Column k=3: A065513.
%C A201685 Row sums:   A001865.
%C A201685 T(n,n) = (n-1)!, T(n,n-1) = n!.
%C A201685 Sum_{k=1..n} T(n,k)*k = n^n. - _Geoffrey Critzer_, May 13 2013
%C A201685 From the asymptotic given by N-E. Fahssi in A001865, we see the expected size of the cycle grows as (2*n/Pi)^(1/2). - _Geoffrey Critzer_, May 13 2013
%C A201685 Central terms: A277168. - _Paul D. Hanna_, Oct 01 2016
%H A201685 Alois P. Heinz, <a href="/A201685/b201685.txt">Rows n = 1..141, flattened</a>
%F A201685 E.g.f.: log(1/(1-y*A(x))) where A(x) is the e.g.f. for A000169.
%F A201685 T(n,k) = binomial(n-1,k-1)*n^(n-k)*(k-1)!. - _Geoffrey Critzer_, May 13 2013
%e A201685 Triangle begins as:
%e A201685      1;
%e A201685      2,    1;
%e A201685      9,    6,    2;
%e A201685     64,   48,   24,    6;
%e A201685    625,  500,  300,  120,  24;
%e A201685   7776, 6480, 4320, 2160, 720, 120;
%p A201685 T:= (n, k)-> binomial(n-1, k-1)*n^(n-k)*(k-1)!:
%p A201685 seq(seq(T(n, k), k=1..n), n=1..12);  # _Alois P. Heinz_, Aug 14 2013
%t A201685 f[list_] := Select[list, # > 0 &]; t = Sum[n^(n - 1) x^n/n!, {n, 1, 20}]; Map[f, Drop[Range[0, 10]! CoefficientList[Series[Log[1/(1 - y t)], {x, 0, 10}], {x, y}], 1]] // Grid
%o A201685 (PARI) T(n,k) = binomial(n-1,k-1)*n^(n-k)*(k-1)!; \\ _G. C. Greubel_, Jan 08 2020
%o A201685 (Magma) [Binomial(n-1,k-1)*n^(n-k)*Factorial(k-1): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Jan 08 2020
%o A201685 (Sage) [[binomial(n-1,k-1)*n^(n-k)*factorial(k-1) for k in (1..n)] for n in (1..12)] # _G. C. Greubel_, Jan 08 2020
%o A201685 (GAP) Flat(List([1..12], n-> List([1..n], k-> Binomial(n-1,k-1)*n^(n-k)*Factorial(k-1) ))); # _G. C. Greubel_, Jan 08 2020
%Y A201685 Cf. A000169, A001865, A053506, A065513, A277168.
%K A201685 nonn,tabl
%O A201685 1,2
%A A201685 _Geoffrey Critzer_, Dec 03 2011