This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A350446 #42 Mar 24 2023 15:50:28
%S A350446 1,1,3,1,16,11,125,128,3,1296,1734,95,16807,27409,2425,15,262144,
%T A350446 499400,61054,945,4782969,10346328,1605534,42280,105,100000000,
%U A350446 240722160,44981292,1706012,11025,2357947691,6222652233,1351343346,67291910,763875,945
%N A350446 Number T(n,k) of endofunctions on [n] with exactly k cycles of length larger than 1; triangle T(n,k), n>=0, 0<=k<=floor(n/2), read by rows.
%H A350446 Alois P. Heinz, <a href="/A350446/b350446.txt">Rows n = 0..200, flattened</a>
%F A350446 From _Mélika Tebni_, Mar 23 2023: (Start)
%F A350446 E.g.f. of column k: (W(-x)-log(1 + W(-x)))^k / (exp(W(-x))*k!), W(x) the Lambert W-function.
%F A350446 T(n,k) = Sum_{j=k..n} n^(n-j)*binomial(n-1,j-1)*A136394(j,k), for n > 0.
%F A350446 T(n,k) = Sum_{j=k..n} (n-j+1)^(n-j-1)*binomial(n,j)*A350452(j,k).
%F A350446 Sum_{k=0..n/2} (k+1)*T(n,k) = A190314(n), for n > 0.
%F A350446 Sum_{k=0..n/2} 2^k*T(n,k) = A217701(n). (End)
%e A350446 Triangle T(n,k) begins:
%e A350446 1;
%e A350446 1;
%e A350446 3, 1;
%e A350446 16, 11;
%e A350446 125, 128, 3;
%e A350446 1296, 1734, 95;
%e A350446 16807, 27409, 2425, 15;
%e A350446 262144, 499400, 61054, 945;
%e A350446 4782969, 10346328, 1605534, 42280, 105;
%e A350446 100000000, 240722160, 44981292, 1706012, 11025;
%e A350446 2357947691, 6222652233, 1351343346, 67291910, 763875, 945;
%e A350446 ...
%p A350446 c:= proc(n) option remember; add(n!*n^(n-k-1)/(n-k)!, k=2..n) end:
%p A350446 t:= proc(n) option remember; n^(n-1) end:
%p A350446 b:= proc(n) option remember; expand(`if`(n=0, 1, add(
%p A350446 b(n-i)*binomial(n-1, i-1)*(c(i)*x+t(i)), i=1..n)))
%p A350446 end:
%p A350446 T:= n-> (p-> seq(coeff(p, x, i), i=0..n/2))(b(n)):
%p A350446 seq(T(n), n=0..12);
%p A350446 # second Maple program:
%p A350446 egf := k-> (LambertW(-x)-log(1+LambertW(-x)))^k/(exp(LambertW(-x))*k!):
%p A350446 A350446 := (n, k)-> n! * coeff(series(egf(k), x, n+1), x, n):
%p A350446 seq(print(seq(A350446(n, k), k=0..n/2)), n=0..10); # _Mélika Tebni_, Mar 23 2023
%t A350446 c[n_] := c[n] = Sum[n!*n^(n - k - 1)/(n - k)!, {k, 2, n}];
%t A350446 t[n_] := t[n] = n^(n - 1);
%t A350446 b[n_] := b[n] = Expand[If[n == 0, 1, Sum[
%t A350446 b[n - i]*Binomial[n - 1, i - 1]*(c[i]*x + t[i]), {i, 1, n}]]];
%t A350446 T[n_] := With[{p = b[n]}, Table[Coefficient[p, x, i], {i, 0, n/2}]];
%t A350446 Table[T[n], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, May 06 2022, after _Alois P. Heinz_ *)
%Y A350446 Column k=0 gives A000272(n+1).
%Y A350446 Row sums give A000312.
%Y A350446 T(2n,n) gives A001147.
%Y A350446 Cf. A055134, A060281, A349454, A350452.
%Y A350446 Cf. A136394, A190314, A217701.
%K A350446 nonn,tabf
%O A350446 0,3
%A A350446 _Alois P. Heinz_, Dec 31 2021