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%I #28 Jan 21 2023 04:29:08
%S 1,0,0,1,0,8,0,78,3,0,944,80,0,13800,1810,15,0,237432,41664,840,0,
%T 4708144,1022252,34300,105,0,105822432,27098784,1286432,10080,0,
%U 2660215680,778128336,47790540,648900,945,0,73983185000,24165049920,1815578160,36048320,138600
%N Number T(n,k) of endofunctions on [n] with exactly k connected components and no fixed points; triangle T(n,k), n>=0, 0<=k<=floor(n/2), read by rows.
%C For k >= 2 and p prime, T(p,k) == 0 (mod 4*p*(p-1)). - _Mélika Tebni_, Jan 20 2023
%H Alois P. Heinz, <a href="/A350452/b350452.txt">Rows n = 0..200, flattened</a>
%F From _Mélika Tebni_, Jan 20 2023: (Start)
%F E.g.f. column k: (LambertW(-x) - log(1 + LambertW(-x)))^k / k!.
%F -Sum_{k=1..n/2} (-1)^k*T(n,k) = A071720(n+1), for n > 0.
%F -Sum_{k=1..n/2} (-1)^k*T(n,k) / (n-1) = A007830(n-2), for n > 1.
%F T(n,k) = Sum_{j=k..n} n^(n-j)*binomial(n-1, j-1)*A106828(j, k) for n > 0. (End)
%e Triangle T(n,k) begins:
%e 1;
%e 0;
%e 0, 1;
%e 0, 8;
%e 0, 78, 3;
%e 0, 944, 80;
%e 0, 13800, 1810, 15;
%e 0, 237432, 41664, 840;
%e 0, 4708144, 1022252, 34300, 105;
%e 0, 105822432, 27098784, 1286432, 10080;
%e 0, 2660215680, 778128336, 47790540, 648900, 945;
%e ...
%p c:= proc(n) option remember; add(n!*n^(n-k-1)/(n-k)!, k=2..n) end:
%p b:= proc(n) option remember; expand(`if`(n=0, 1, add(
%p b(n-i)*binomial(n-1, i-1)*x*c(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);
%t c[n_] := c[n] = Sum[n!*n^(n - k - 1)/(n - k)!, {k, 2, n}];
%t b[n_] := b[n] = Expand[If[n == 0, 1, Sum[
%t b[n - i]*Binomial[n - 1, i - 1]*x*c[i], {i, 1, n}]]];
%t T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n/2}]][b[n]];
%t Table[T[n], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Mar 18 2022, after _Alois P. Heinz_ *)
%o (PARI) \\ here AS1(n,k) gives associated Stirling numbers of 1st kind.
%o AS1(n,k)={(-1)^(n+k)*sum(i=0, k, (-1)^i * binomial(n, i) * stirling(n-i, k-i, 1) )}
%o T(n,k) = {if(n==0, k==0, sum(j=k, n, n^(n-j)*binomial(n-1, j-1)*AS1(j,k)))} \\ _Andrew Howroyd_, Jan 20 2023
%Y Columns k=0-1 give: A000007, A000435.
%Y Row sums give A065440.
%Y T(2n,n) gives A001147.
%Y Cf. A060281, A061356, A350446.
%Y Cf. A071720, A007830, A106828.
%K nonn,tabf
%O 0,6
%A _Alois P. Heinz_, Dec 31 2021
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