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A350212 Number T(n,k) of endofunctions on [n] with exactly k isolated fixed points; triangle T(n,k), n >= 0, 0 <= k <= n, read by rows.

%I A350212 #43 Dec 05 2022 12:40:17
%S A350212 1,0,1,3,0,1,17,9,0,1,169,68,18,0,1,2079,845,170,30,0,1,31261,12474,
%T A350212 2535,340,45,0,1,554483,218827,43659,5915,595,63,0,1,11336753,4435864,
%U A350212 875308,116424,11830,952,84,0,1,262517615,102030777,19961388,2625924,261954,21294,1428,108,0,1
%N A350212 Number T(n,k) of endofunctions on [n] with exactly k isolated fixed points; triangle T(n,k), n >= 0, 0 <= k <= n, read by rows.
%H A350212 Alois P. Heinz, <a href="/A350212/b350212.txt">Rows n = 0..140, flattened</a>
%F A350212 Sum_{k=0..n} k * T(n,k) = A055897(n).
%F A350212 Sum_{k=1..n} T(n,k) = A350134(n).
%F A350212 From _Mélika Tebni_, Nov 24 2022: (Start)
%F A350212 T(n,k) = binomial(n, k)*|A069856(n-k)|.
%F A350212 E.g.f. column k: exp(-x)*x^k / ((1 + LambertW(-x))*k!).
%F A350212 T(n,k) = Sum_{j=0..n} (-1)^(j-k)*binomial(j, k)*binomial(n, j)*(n-j)^(n-j). (End)
%e A350212 T(3,1) = 9: 122, 133, 132, 121, 323, 321, 113, 223, 213.
%e A350212 Triangle T(n,k) begins:
%e A350212          1;
%e A350212          0,       1;
%e A350212          3,       0,      1;
%e A350212         17,       9,      0,      1;
%e A350212        169,      68,     18,      0,     1;
%e A350212       2079,     845,    170,     30,     0,   1;
%e A350212      31261,   12474,   2535,    340,    45,   0,  1;
%e A350212     554483,  218827,  43659,   5915,   595,  63,  0, 1;
%e A350212   11336753, 4435864, 875308, 116424, 11830, 952, 84, 0, 1;
%e A350212   ...
%p A350212 g:= proc(n) option remember; add(n^(n-j)*(n-1)!/(n-j)!, j=1..n) end:
%p A350212 b:= proc(n, m) option remember; `if`(n=0, x^m, add(g(i)*
%p A350212       b(n-i, m+`if`(i=1, 1, 0))*binomial(n-1, i-1), i=1..n))
%p A350212     end:
%p A350212 T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 0)):
%p A350212 seq(T(n), n=0..10);
%p A350212 # second Maple program:
%p A350212 A350212 := (n,k)-> add((-1)^(j-k)*binomial(j,k)*binomial(n,j)*(n-j)^(n-j), j=0..n):
%p A350212 seq(print(seq(A350212(n, k), k=0..n)), n=0..9); # _Mélika Tebni_, Nov 24 2022
%t A350212 g[n_] := g[n] = Sum[n^(n - j)*(n - 1)!/(n - j)!, {j, 1, n}];
%t A350212 b[n_, m_] := b[n, m] = If[n == 0, x^m, Sum[g[i]*
%t A350212      b[n - i, m + If[i == 1, 1, 0]]*Binomial[n - 1, i - 1], {i, 1, n}]];
%t A350212 T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, 0]];
%t A350212 Table[T[n], {n, 0, 10}] // Flatten (* _Jean-François Alcover_, Mar 11 2022, after _Alois P. Heinz_ *)
%Y A350212 Columns k=0-1 give: |A069856|, A348590.
%Y A350212 Row sums give A000312.
%Y A350212 T(n+1,n-1) gives A045943.
%Y A350212 Cf. A001865, A008290, A008291, A055134, A055897, A060281, A086659, A124323, A350134, A349454.
%K A350212 nonn,tabl
%O A350212 0,4
%A A350212 _Alois P. Heinz_, Dec 19 2021