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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.

%I A350454 #22 Dec 22 2022 10:15:10
%S A350454 1,0,1,2,8,9,81,76,12,1024,875,180,15625,12606,2910,120,279936,217217,
%T A350454 53550,3780,5764801,4348856,1118936,102480,1680,134217728,99111735,
%U A350454 26280072,2817360,90720,3486784401,2532027610,686569050,81864720,3729600,30240
%N 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.
%H A350454 Alois P. Heinz, <a href="/A350454/b350454.txt">Rows n = 0..200, flattened</a>
%F A350454 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
%F A350454 From _Mélika Tebni_, Dec 22 2022: (Start)
%F A350454 For n > 1, T(n,1) = n*A045531(n-1).
%F A350454 Sum_{k=0..n} (-1)^(n-k)*T(n+k,k) = 2^n.
%F A350454 Sum_{k=0..n} (-1)^(n-k)*T(n+k,k)/(n+k-1) = 1/n, for n > 1. (End)
%e A350454 Triangle T(n,k) begins:
%e A350454            1;
%e A350454            0;
%e A350454            1,          2;
%e A350454            8,          9;
%e A350454           81,         76,        12;
%e A350454         1024,        875,       180;
%e A350454        15625,      12606,      2910,      120;
%e A350454       279936,     217217,     53550,     3780;
%e A350454      5764801,    4348856,   1118936,   102480,    1680;
%e A350454    134217728,   99111735,  26280072,  2817360,   90720;
%e A350454   3486784401, 2532027610, 686569050, 81864720, 3729600, 30240;
%e A350454   ...
%p A350454 c:= proc(n) option remember; add(n!*n^(n-k-1)/(n-k)!, k=2..n) end:
%p A350454 t:= proc(n) option remember; n^(n-1) end:
%p A350454 b:= proc(n) option remember; expand(`if`(n=0, 1, add(b(n-i)*
%p A350454       binomial(n-1, i-1)*(c(i)+`if`(i=1, 0, x*t(i))), i=1..n)))
%p A350454     end:
%p A350454 T:= n-> (p-> seq(coeff(p, x, i), i=0..n/2))(b(n)):
%p A350454 seq(T(n), n=0..12);
%p A350454 # second Maple program:
%p A350454 egf := k-> exp(LambertW(-x))*(-x-LambertW(-x))^k/((1+LambertW(-x))*k!):
%p A350454 A350454 := (n, k)-> n! * coeff(series(egf(k), x, n+1), x, n):
%p A350454 seq(print(seq(A350454(n, k), k=0..n/2)), n=0..9); # _Mélika Tebni_, Nov 22 2022
%t A350454 c[n_] := c[n] = Sum[n!*n^(n - k - 1)/(n - k)!, {k, 2, n}];
%t A350454 t[n_] := t[n] = n^(n - 1);
%t A350454 b[n_] := b[n] = Expand[If[n == 0, 1, Sum[b[n - i]*
%t A350454      Binomial[n - 1, i - 1]*(c[i] + If[i == 1, 0, x*t[i]]), {i, 1, n}]]];
%t A350454 T[n_] := With[{p = b[n]}, Table[Coefficient[p, x, i], {i, 0, n/2}]];
%t A350454 Table[T[n], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, May 06 2022, after _Alois P. Heinz_ *)
%Y A350454 Column k=0 gives A065440.
%Y A350454 Row sums give |A069856|.
%Y A350454 T(2n,n) gives A001813.
%Y A350454 Cf. A349454.
%K A350454 nonn,tabf
%O A350454 0,4
%A A350454 _Alois P. Heinz_, Dec 31 2021