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.
1, 1, 3, 1, 16, 11, 125, 128, 3, 1296, 1734, 95, 16807, 27409, 2425, 15, 262144, 499400, 61054, 945, 4782969, 10346328, 1605534, 42280, 105, 100000000, 240722160, 44981292, 1706012, 11025, 2357947691, 6222652233, 1351343346, 67291910, 763875, 945
Offset: 0
Examples
Triangle T(n,k) begins:
1;
1;
3, 1;
16, 11;
125, 128, 3;
1296, 1734, 95;
16807, 27409, 2425, 15;
262144, 499400, 61054, 945;
4782969, 10346328, 1605534, 42280, 105;
100000000, 240722160, 44981292, 1706012, 11025;
2357947691, 6222652233, 1351343346, 67291910, 763875, 945;
...
Links
- Alois P. Heinz, Rows n = 0..200, flattened
Crossrefs
Programs
-
Maple
c:= proc(n) option remember; add(n!*n^(n-k-1)/(n-k)!, k=2..n) end: t:= proc(n) option remember; n^(n-1) end: b:= proc(n) option remember; expand(`if`(n=0, 1, add( b(n-i)*binomial(n-1, i-1)*(c(i)*x+t(i)), i=1..n))) end: T:= n-> (p-> seq(coeff(p, x, i), i=0..n/2))(b(n)): seq(T(n), n=0..12); # second Maple program: egf := k-> (LambertW(-x)-log(1+LambertW(-x)))^k/(exp(LambertW(-x))*k!): A350446 := (n, k)-> n! * coeff(series(egf(k), x, n+1), x, n): seq(print(seq(A350446(n, k), k=0..n/2)), n=0..10); # Mélika Tebni, Mar 23 2023 -
Mathematica
c[n_] := c[n] = Sum[n!*n^(n - k - 1)/(n - k)!, {k, 2, n}]; t[n_] := t[n] = n^(n - 1); b[n_] := b[n] = Expand[If[n == 0, 1, Sum[ b[n - i]*Binomial[n - 1, i - 1]*(c[i]*x + t[i]), {i, 1, n}]]]; T[n_] := With[{p = b[n]}, Table[Coefficient[p, x, i], {i, 0, n/2}]]; Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, May 06 2022, after Alois P. Heinz *)
Formula
From Mélika Tebni, Mar 23 2023: (Start)
E.g.f. of column k: (W(-x)-log(1 + W(-x)))^k / (exp(W(-x))*k!), W(x) the Lambert W-function.
T(n,k) = Sum_{j=k..n} n^(n-j)*binomial(n-1,j-1)*A136394(j,k), for n > 0.
T(n,k) = Sum_{j=k..n} (n-j+1)^(n-j-1)*binomial(n,j)*A350452(j,k).
Sum_{k=0..n/2} (k+1)*T(n,k) = A190314(n), for n > 0.
Sum_{k=0..n/2} 2^k*T(n,k) = A217701(n). (End)
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