A350157 Total number of nodes in the smallest connected component summed over all endofunctions on [n].
0, 1, 7, 61, 709, 9911, 167111, 3237921, 71850913, 1780353439, 49100614399, 1482061739423, 48873720208853, 1740252983702871, 66793644836081827, 2740470162691675711, 120029057782404141841, 5575505641199441262767, 274412698693082818767335, 14236421024010426118259883
Offset: 0
Keywords
Examples
a(2) = 7 = 2 + 2 + 1 + 2: 11, 22, 12, 21.
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..385
Programs
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Maple
g:= proc(n) option remember; add(n^(n-j)*(n-1)!/(n-j)!, j=1..n) end: b:= proc(n, m) option remember; `if`(n=0, x^m, add( b(n-i, min(m, i))*g(i)*binomial(n-1, i-1), i=1..n)) end: a:= n-> (p-> add(coeff(p, x, i)*i, i=0..n))(b(n,n)): seq(a(n), n=0..23); -
Mathematica
g[n_] := g[n] = Sum[n^(n - j)*(n - 1)!/(n - j)!, {j, 1, n}]; b[n_, m_] := b[n, m] = If[n == 0, x^m, Sum[ b[n - i, Min[m, i]]*g[i]*Binomial[n - 1, i - 1], {i, 1, n}]]; a[n_] := Function[p, Sum[Coefficient[p, x, i]*i, {i, 0, n}]][b[n, n]]; Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Apr 27 2022, after Alois P. Heinz *)
Formula
a(n) = Sum_{k=1..n} k * A347999(n,k).