A350454 Number T(n,k) of endofunctions on [n] with exactly k fixed points, none of which are isolated; triangle T(n,k), n >= 0, 0 <= k <= n/2, read by rows.
1, 0, 1, 2, 8, 9, 81, 76, 12, 1024, 875, 180, 15625, 12606, 2910, 120, 279936, 217217, 53550, 3780, 5764801, 4348856, 1118936, 102480, 1680, 134217728, 99111735, 26280072, 2817360, 90720, 3486784401, 2532027610, 686569050, 81864720, 3729600, 30240
Offset: 0
Examples
Triangle T(n,k) begins:
1;
0;
1, 2;
8, 9;
81, 76, 12;
1024, 875, 180;
15625, 12606, 2910, 120;
279936, 217217, 53550, 3780;
5764801, 4348856, 1118936, 102480, 1680;
134217728, 99111735, 26280072, 2817360, 90720;
3486784401, 2532027610, 686569050, 81864720, 3729600, 30240;
...
Links
- Alois P. Heinz, Rows n = 0..200, flattened
Programs
-
Maple
c:= proc(n) option remember; add(n!*n^(n-k-1)/(n-k)!, k=2..n) end: t:= proc(n) option remember; n^(n-1) end: b:= proc(n) option remember; expand(`if`(n=0, 1, add(b(n-i)* binomial(n-1, i-1)*(c(i)+`if`(i=1, 0, x*t(i))), i=1..n))) end: T:= n-> (p-> seq(coeff(p, x, i), i=0..n/2))(b(n)): seq(T(n), n=0..12); # second Maple program: egf := k-> exp(LambertW(-x))*(-x-LambertW(-x))^k/((1+LambertW(-x))*k!): A350454 := (n, k)-> n! * coeff(series(egf(k), x, n+1), x, n): seq(print(seq(A350454(n, k), k=0..n/2)), n=0..9); # Mélika Tebni, Nov 22 2022 -
Mathematica
c[n_] := c[n] = Sum[n!*n^(n - k - 1)/(n - k)!, {k, 2, n}]; t[n_] := t[n] = n^(n - 1); b[n_] := b[n] = Expand[If[n == 0, 1, Sum[b[n - i]* Binomial[n - 1, i - 1]*(c[i] + If[i == 1, 0, x*t[i]]), {i, 1, n}]]]; T[n_] := With[{p = b[n]}, Table[Coefficient[p, x, i], {i, 0, n/2}]]; Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, May 06 2022, after Alois P. Heinz *)
Formula
E.g.f. column k: exp(W(-x))*(-x - W(-x))^k / ((1 + W(-x))*k!), W(x) the Lambert W-function. - Mélika Tebni, Nov 22 2022
From Mélika Tebni, Dec 22 2022: (Start)
For n > 1, T(n,1) = n*A045531(n-1).
Sum_{k=0..n} (-1)^(n-k)*T(n+k,k) = 2^n.
Sum_{k=0..n} (-1)^(n-k)*T(n+k,k)/(n+k-1) = 1/n, for n > 1. (End)