This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A066324 #53 Dec 06 2024 08:45:41
%S A066324 1,2,2,9,12,6,64,96,72,24,625,1000,900,480,120,7776,12960,12960,8640,
%T A066324 3600,720,117649,201684,216090,164640,88200,30240,5040,2097152,
%U A066324 3670016,4128768,3440640,2150400,967680,282240,40320,43046721
%N A066324 Number of endofunctions on n labeled points constructed from k rooted trees.
%C A066324 T(n,k) = number of endofunctions with k recurrent elements. - _Mitch Harris_, Jul 06 2006
%C A066324 The sum of row n is n^n, for any n. Basically the same sequence arises when studying random mappings (see A243203, A243202). - _Stanislav Sykora_, Jun 01 2014
%D A066324 F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 87, see (2.3.28).
%D A066324 I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983, ex. 3.3.32.
%H A066324 Alois P. Heinz, <a href="/A066324/b066324.txt">Rows n = 1..141, flattened</a>
%F A066324 T(n,k) = k*n^(n-k)*(n-1)!/(n-k)!.
%F A066324 E.g.f. (relative to x): A(x, y)=1/(1-y*B(x)) - 1 = y*x +(2*y+2*y^2)*x^2/2! + (9*y+12*y^2+6*y^3)*x^3/3! + ..., where B(x) is e.g.f. A000169.
%F A066324 From _Peter Bala_, Sep 30 2011: (Start)
%F A066324 Let F(x,t) = x/(1+t*x)*exp(-x/(1+t*x)) = x*(1 - (1+t)*x + (1+4*t+2*t^2)*x^2/2! - ...). F is essentially the e.g.f. for A144084 (see also A021010). Then the e.g.f. for the present table is t*F(x,t)^(-1), where the compositional inverse is taken with respect to x.
%F A066324 Removing a factor of n from the n-th row entries results in A122525 in row reversed form.
%F A066324 (End)
%F A066324 Sum_{k=2..n} (k-1) * T(n,k) = A001864(n). - _Geoffrey Critzer_, Aug 19 2013
%F A066324 Sum_{k=1..n} k * T(n,k) = A063169(n). - _Alois P. Heinz_, Dec 15 2021
%e A066324 Triangle T(n,k) begins:
%e A066324 1;
%e A066324 2, 2;
%e A066324 9, 12, 6;
%e A066324 64, 96, 72, 24;
%e A066324 625, 1000, 900, 480, 120;
%e A066324 7776, 12960, 12960, 8640, 3600, 720;
%e A066324 117649, 201684, 216090, 164640, 88200, 30240, 5040;
%e A066324 ...
%p A066324 T:= (n, k)-> k*n^(n-k)*(n-1)!/(n-k)!:
%p A066324 seq(seq(T(n, k), k=1..n), n=1..10); # _Alois P. Heinz_, Aug 22 2012
%t A066324 f[list_] := Select[list, # > 0 &]; t = Sum[n^(n - 1) x^n/n!, {n, 1, 20}]; Flatten[Map[f, Drop[Range[0, 10]! CoefficientList[Series[1/(1 - y*t), {x, 0, 10}], {x, y}], 1]]] (* _Geoffrey Critzer_, Dec 05 2011 *)
%o A066324 (PARI) T(n, k)=k*n^(n-k)*(n-1)!/(n-k)! \\ _Charles R Greathouse IV_, Dec 05 2011
%Y A066324 Column 1: A000169.
%Y A066324 Main diagonal: A000142.
%Y A066324 T(n, n-1): A062119.
%Y A066324 Row sums give A000312.
%Y A066324 Cf. A001864, A021010, A063169, A122525, A144084, A243203.
%K A066324 nonn,tabl
%O A066324 1,2
%A A066324 _Christian G. Bower_, Dec 14 2001