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