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A349454 Number T(n,k) of endofunctions on [n] with exactly k fixed points, all of which are isolated; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%I A349454 #59 Apr 02 2023 08:16:35
%S A349454 1,0,1,1,0,1,8,3,0,1,81,32,6,0,1,1024,405,80,10,0,1,15625,6144,1215,
%T A349454 160,15,0,1,279936,109375,21504,2835,280,21,0,1,5764801,2239488,
%U A349454 437500,57344,5670,448,28,0,1,134217728,51883209,10077696,1312500,129024,10206,672,36,0,1
%N A349454 Number T(n,k) of endofunctions on [n] with exactly k fixed points, all of which are isolated; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%H A349454 Alois P. Heinz, <a href="/A349454/b349454.txt">Rows n = 0..140, flattened</a>
%F A349454 T(n,k) = binomial(n,k) * (n-k-1)^(n-k).
%F A349454 From _Mélika Tebni_, Apr 02 2023: (Start)
%F A349454 E.g.f. of column k: -x / (LambertW(-x)*(1+LambertW(-x)))*x^k / k!.
%F A349454 Sum_{k=0..n} k^k*T(n,k) = A217701(n). (End)
%e A349454 Triangle T(n,k) begins:
%e A349454         1;
%e A349454         0,       1;
%e A349454         1,       0,      1;
%e A349454         8,       3,      0,     1;
%e A349454        81,      32,      6,     0,    1;
%e A349454      1024,     405,     80,    10,    0,   1;
%e A349454     15625,    6144,   1215,   160,   15,   0,  1;
%e A349454    279936,  109375,  21504,  2835,  280,  21,  0, 1;
%e A349454   5764801, 2239488, 437500, 57344, 5670, 448, 28, 0, 1;
%e A349454   ...
%p A349454 T:= (n, k)-> binomial(n, k)*(n-k-1)^(n-k):
%p A349454 seq(seq(T(n, k), k=0..n), n=0..10);
%Y A349454 Column k=0 gives A065440.
%Y A349454 Row sums give A204042.
%Y A349454 Main diagonal and first lower diagonal give A000012, A000004.
%Y A349454 T(n+1,n-1) gives A000217.
%Y A349454 T(n+3,n) gives A130809.
%Y A349454 T(n+3,n-1) gives A102741 for n>=1.
%Y A349454 Cf. A055134, A217701, A350212, A350454.
%K A349454 nonn,tabl
%O A349454 0,7
%A A349454 _Alois P. Heinz_, Dec 30 2021