login
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.
3

%I #22 Dec 22 2022 10:15:10

%S 1,0,1,2,8,9,81,76,12,1024,875,180,15625,12606,2910,120,279936,217217,

%T 53550,3780,5764801,4348856,1118936,102480,1680,134217728,99111735,

%U 26280072,2817360,90720,3486784401,2532027610,686569050,81864720,3729600,30240

%N 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 Alois P. Heinz, <a href="/A350454/b350454.txt">Rows n = 0..200, flattened</a>

%F 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 From _Mélika Tebni_, Dec 22 2022: (Start)

%F For n > 1, T(n,1) = n*A045531(n-1).

%F Sum_{k=0..n} (-1)^(n-k)*T(n+k,k) = 2^n.

%F Sum_{k=0..n} (-1)^(n-k)*T(n+k,k)/(n+k-1) = 1/n, for n > 1. (End)

%e Triangle T(n,k) begins:

%e 1;

%e 0;

%e 1, 2;

%e 8, 9;

%e 81, 76, 12;

%e 1024, 875, 180;

%e 15625, 12606, 2910, 120;

%e 279936, 217217, 53550, 3780;

%e 5764801, 4348856, 1118936, 102480, 1680;

%e 134217728, 99111735, 26280072, 2817360, 90720;

%e 3486784401, 2532027610, 686569050, 81864720, 3729600, 30240;

%e ...

%p c:= proc(n) option remember; add(n!*n^(n-k-1)/(n-k)!, k=2..n) end:

%p t:= proc(n) option remember; n^(n-1) end:

%p b:= proc(n) option remember; expand(`if`(n=0, 1, add(b(n-i)*

%p binomial(n-1, i-1)*(c(i)+`if`(i=1, 0, x*t(i))), i=1..n)))

%p end:

%p T:= n-> (p-> seq(coeff(p, x, i), i=0..n/2))(b(n)):

%p seq(T(n), n=0..12);

%p # second Maple program:

%p egf := k-> exp(LambertW(-x))*(-x-LambertW(-x))^k/((1+LambertW(-x))*k!):

%p A350454 := (n, k)-> n! * coeff(series(egf(k), x, n+1), x, n):

%p seq(print(seq(A350454(n, k), k=0..n/2)), n=0..9); # _Mélika Tebni_, Nov 22 2022

%t c[n_] := c[n] = Sum[n!*n^(n - k - 1)/(n - k)!, {k, 2, n}];

%t t[n_] := t[n] = n^(n - 1);

%t b[n_] := b[n] = Expand[If[n == 0, 1, Sum[b[n - i]*

%t Binomial[n - 1, i - 1]*(c[i] + If[i == 1, 0, x*t[i]]), {i, 1, n}]]];

%t T[n_] := With[{p = b[n]}, Table[Coefficient[p, x, i], {i, 0, n/2}]];

%t Table[T[n], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, May 06 2022, after _Alois P. Heinz_ *)

%Y Column k=0 gives A065440.

%Y Row sums give |A069856|.

%Y T(2n,n) gives A001813.

%Y Cf. A349454.

%K nonn,tabf

%O 0,4

%A _Alois P. Heinz_, Dec 31 2021