A001865 -id:A001865 - cp's OEIS frontend

A350079 Triangle read by rows: T(n,k) is the number of endofunctions on [n] whose second-smallest component has size exactly k; n >= 0, 0 <= k <= max(0,n-1).

1, 1, 3, 1, 17, 1, 9, 142, 19, 27, 68, 1569, 201, 135, 510, 710, 21576, 2921, 3465, 2890, 6390, 9414, 355081, 50233, 63630, 20230, 84490, 98847, 151032, 6805296, 1004599, 1196181, 918680, 705740, 1493688, 1812384, 2840648, 148869153, 22872097, 26904339, 23943752, 6351660, 28072548, 30810528, 38348748, 61247664

Offset: 0

Steven Finch, Dec 12 2021

An endofunction on [n] is a function from {1,2,...,n} to {1,2,...,n}.

If the mapping has no second component, then its second-smallest component is defined to have size 0.

			Triangle begins:
       1;
       1;
       3,     1;
      17,     1,     9;
     142,    19,    27,    68;
    1569,   201,   135,   510,   710;
   21576,  2921,  3465,  2890,  6390,  9414;
  355081, 50233, 63630, 20230, 84490, 98847, 151032;
  ...

Alois P. Heinz, Rows n = 0..141, flattened
Steven Finch, Second best, Third worst, Fourth in line, arxiv:2202.07621 [math.CO], 2022.

Column 0 gives gives 1 together with A001865.
Row sums give A000312.
Cf. A001865, A350078, A350080, A350081, A350275, A350276

g:= proc(n) option remember; add(n^(n-j)*(n-1)!/(n-j)!, j=1..n) end:
b:= proc(n, l) option remember; `if`(n=0, x^subs(infinity=0, l)[2],
      add(b(n-i, sort([l[], i])[1..2])*g(i)*binomial(n-1, i-1), i=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [infinity$2])):
seq(T(n), n=0..12);  # Alois P. Heinz, Dec 17 2021

g[n_] := g[n] = Sum[n^(n - j)*(n - 1)!/(n - j)!, {j, 1, n}];
b[n_, l_] := b[n, l] = If[n == 0, x^(l /. Infinity -> 0)[[2]], Sum[b[n - i, Sort[Append[l, i]][[1;;2]]]*g[i]*Binomial[n - 1, i - 1], {i, 1, n}]];
T[n_] := With[{p = b[n, {Infinity, Infinity}]}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 28 2021, after Alois P. Heinz *)

More terms (two rows) from Alois P. Heinz, Dec 15 2021

A350080 Irregular triangle read by rows: T(n,k) is the number of endofunctions on [n] whose third-largest component has size exactly k; n >= 0, 0 <= k <= floor(n/3).

Original entry on oeis.org

1, 1, 4, 26, 1, 237, 19, 2789, 336, 40270, 5981, 405, 689450, 115193, 18900, 13657756, 2459955, 659505, 307348641, 58366045, 20330163, 1375640, 7745565616, 1530739594, 623758590, 99936200, 216114310994, 44076571672, 19795671225, 5325116720

Offset: 0

Steven Finch, Dec 12 2021

An endofunction on [n] is a function from {1,2,...,n} to {1,2,...,n}.

If the mapping has no third component, then its third-largest component is defined to have size 0.

			Triangle begins:
       1;
       1;
       4;
      26,     1;
     237,    19;
    2789,   336;
   40270,   5981,   405;
  689450, 115193, 18900;
  ...

Alois P. Heinz, Rows n = 0..120, flattened
Steven Finch, Second best, Third worst, Fourth in line, arxiv:2202.07621 [math.CO], 2022.

Row sums give A000312.

Cf. A001865, A350078, A350079, A350081, A350275, A350276.

g:= proc(n) option remember; add(n^(n-j)*(n-1)!/(n-j)!, j=1..n) end:
b:= proc(n, l) option remember; `if`(n=0, x^l[1], add(g(i)*
      b(n-i, sort([l[], i])[-3..-1])*binomial(n-1, i-1), i=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [0$3])):
seq(T(n), n=0..12);  # Alois P. Heinz, Dec 17 2021

g[n_] := g[n] = Sum[n^(n - j)*(n - 1)!/(n - j)!, {j, 1, n}];
b[n_, l_] := b[n, l] = If[n == 0, x^l[[1]], Sum[g[i]*b[n - i, Sort[ Append[l, i]][[-3 ;; -1]]]*Binomial[n - 1, i - 1], {i, 1, n}]];
T[n_] := With[{p = b[n, {0, 0, 0}]}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 28 2021, after Alois P. Heinz *)

More terms (4 rows) from Alois P. Heinz, Dec 16 2021

A350081 Triangle read by rows: T(n,k) is the number of endofunctions on [n] whose third-smallest component has size exactly k; n >= 0, 0 <= k <= max(0,n-2).

Original entry on oeis.org

1, 1, 4, 26, 1, 237, 1, 18, 2789, 31, 135, 170, 40270, 386, 810, 3060, 2130, 689450, 6574, 13545, 36295, 44730, 32949, 13657756, 129291, 327285, 323680, 944300, 790776, 604128, 307348641, 2910709, 7207137, 6602120, 15476580, 18780930, 16311456, 12782916

Offset: 0

Steven Finch, Dec 12 2021

An endofunction on [n] is a function from {1,2,...,n} to {1,2,...,n}.

If the mapping has no third component, then its third-smallest component is defined to have size 0.

			Triangle begins:
       1;
       1;
       4;
      26,    1;
     237,    1,    18;
    2789,   31,   135,   170;
   40270,  386,   810,  3060,  2130;
  689450, 6574, 13545, 36295, 44730, 32949;
  ...

Alois P. Heinz, Rows n = 0..120, flattened
Steven Finch, Second best, Third worst, Fourth in line, arxiv:2202.07621 [math.CO], 2022.

Row sums give A000312.

Cf. A001865, A350078, A350079, A350080, A350275, A350276.

g:= proc(n) option remember; add(n^(n-j)*(n-1)!/(n-j)!, j=1..n) end:
b:= proc(n, l) option remember; `if`(n=0, x^subs(infinity=0, l)[3],
      add(b(n-i, sort([l[], i])[1..3])*g(i)*binomial(n-1, i-1), i=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [infinity$3])):
seq(T(n), n=0..12);  # Alois P. Heinz, Dec 17 2021

g[n_] := g[n] = Sum[n^(n - j)*(n - 1)!/(n - j)!, {j, 1, n}];
b[n_, l_] := b[n, l] = If[n == 0, x^(l /. Infinity -> 0)[[3]], Sum[b[n - i, Sort[Append[l, i]][[1 ;; 3]]]*g[i]*Binomial[n - 1, i - 1], {i, 1, n}]];
T[n_] := With[{p = b[n, {Infinity, Infinity, Infinity}]}, Table[ Coefficient[p, x, i], {i, 0, Exponent[p, x]}]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 28 2021, after Alois P. Heinz *)

More terms (two rows) from Alois P. Heinz, Dec 16 2021

A350275 Irregular triangle read by rows: T(n,k) is the number of endofunctions on [n] whose fourth-largest component has size exactly k; n >= 0, 0 <= k <= floor(n/4).

Original entry on oeis.org

1, 1, 4, 27, 255, 1, 3094, 31, 45865, 791, 803424, 20119, 16239720, 528991, 8505, 372076163, 14689441, 654885, 9529560676, 435580164, 34859160, 269819334245, 13846282341, 1646054025, 8369112382488, 471890017358, 73811825010, 1286223400

Offset: 0

Steven Finch, Dec 22 2021

An endofunction on [n] is a function from {1,2,...,n} to {1,2,...,n}.

If the mapping has no fourth component, then its fourth-largest component is defined to have size 0.

			Triangle begins:
       1;
       1;
       4;
      27;
     255,     1;
    3094,    31;
   45865,   791;
  803424, 20119;
  ...

Alois P. Heinz, Rows n = 0..100, flattened
Steven Finch, Second best, Third worst, Fourth in line, arxiv:2202.07621 [math.CO], 2022.

Row sums give A000312.

Cf. A001865, A350078, A350079, A350080, A350081, A350276.

g:= proc(n) option remember; add(n^(n-j)*(n-1)!/(n-j)!, j=1..n) end:
b:= proc(n, l) option remember; `if`(n=0, x^l[1], add(g(i)*
      b(n-i, sort([l[], i])[-4..-1])*binomial(n-1, i-1), i=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [0$4])):
seq(T(n), n=0..14);  # Alois P. Heinz, Dec 22 2021

g[n_] := g[n] = Sum[n^(n - j)*(n - 1)!/(n - j)!, {j, 1, n}];
b[n_, l_] := b[n, l] = If[n == 0, x^l[[1]], Sum[g[i]*b[n - i, Sort[ Append[l, i]][[-4 ;; -1]]]*Binomial[n - 1, i - 1], {i, 1, n}]];
T[n_] := With[{p = b[n, {0, 0, 0, 0}]}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]];
Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Dec 28 2021, after Alois P. Heinz *)

A350276 Irregular triangle read by rows: T(n,k) is the number of endofunctions on [n] whose fourth-smallest component has size exactly k; n >= 0, 0 <= k <= max(0,n-3).

Original entry on oeis.org

1, 1, 4, 27, 255, 1, 3094, 1, 30, 45865, 46, 405, 340, 803424, 659, 3780, 10710, 4970, 16239720, 12867, 48405, 209440, 178920, 87864, 372076163, 284785, 1225665, 3005940, 5457060, 3558492, 1812384, 9529560676, 7126384, 32262300, 51205700, 135084600, 120593340, 81557280, 42609720

Offset: 0

Steven Finch, Dec 22 2021

An endofunction on [n] is a function from {1,2,...,n} to {1,2,...,n}.

If the mapping has no fourth component, then its fourth-smallest component is defined to have size 0.

			Triangle begins:
       1;
       1;
       4;
      27;
     255,   1;
    3094,   1,   30;
   45865,  46,  405,   340;
  803424, 659, 3780, 10710, 4970;
  ...

Alois P. Heinz, Rows n = 0..100, flattened
Steven Finch, Second best, Third worst, Fourth in line, arxiv:2202.07621 [math.CO], 2022.

Row sums give A000312.

Cf. A001865, A350078, A350079, A350080, A350081, A350275.

g:= proc(n) option remember; add(n^(n-j)*(n-1)!/(n-j)!, j=1..n) end:
b:= proc(n, l) option remember; `if`(n=0, x^subs(infinity=0, l)[4],
      add(b(n-i, sort([l[], i])[1..4])*g(i)*binomial(n-1, i-1), i=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [infinity$4])):
seq(T(n), n=0..12);  # Alois P. Heinz, Dec 22 2021

g[n_] := g[n] = Sum[n^(n - j)*(n - 1)!/(n - j)!, {j, 1, n}];
b[n_, l_] := b[n, l] = If[n == 0, x^(l /. Infinity -> 0)[[4]], Sum[b[n - i, Sort[Append[l, i]][[1 ;; 4]]]*g[i]*Binomial[n - 1, i - 1], {i, 1, n}]];
T[n_] := With[{p = b[n, Table[Infinity, {4}]]}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 28 2021, after Alois P. Heinz *)

A350212 Number T(n,k) of endofunctions on [n] with exactly k isolated fixed points; triangle T(n,k), n >= 0, 0 <= k <= n, read by rows.

Original entry on oeis.org

1, 0, 1, 3, 0, 1, 17, 9, 0, 1, 169, 68, 18, 0, 1, 2079, 845, 170, 30, 0, 1, 31261, 12474, 2535, 340, 45, 0, 1, 554483, 218827, 43659, 5915, 595, 63, 0, 1, 11336753, 4435864, 875308, 116424, 11830, 952, 84, 0, 1, 262517615, 102030777, 19961388, 2625924, 261954, 21294, 1428, 108, 0, 1

Offset: 0

Alois P. Heinz, Dec 19 2021

			T(3,1) = 9: 122, 133, 132, 121, 323, 321, 113, 223, 213.
Triangle T(n,k) begins:
         1;
         0,       1;
         3,       0,      1;
        17,       9,      0,      1;
       169,      68,     18,      0,     1;
      2079,     845,    170,     30,     0,   1;
     31261,   12474,   2535,    340,    45,   0,  1;
    554483,  218827,  43659,   5915,   595,  63,  0, 1;
  11336753, 4435864, 875308, 116424, 11830, 952, 84, 0, 1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

Columns k=0-1 give: |A069856|, A348590.
Row sums give A000312.
T(n+1,n-1) gives A045943.
Cf. A001865, A008290, A008291, A055134, A055897, A060281, A086659, A124323, A350134, A349454.

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(g(i)*
      b(n-i, m+`if`(i=1, 1, 0))*binomial(n-1, i-1), i=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 0)):
seq(T(n), n=0..10);
# second Maple program:
A350212 := (n,k)-> add((-1)^(j-k)*binomial(j,k)*binomial(n,j)*(n-j)^(n-j), j=0..n):
seq(print(seq(A350212(n, k), k=0..n)), n=0..9); # Mélika Tebni, Nov 24 2022

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[g[i]*
     b[n - i, m + If[i == 1, 1, 0]]*Binomial[n - 1, i - 1], {i, 1, n}]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, 0]];
Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Mar 11 2022, after Alois P. Heinz *)

Sum_{k=0..n} k * T(n,k) = A055897(n).
Sum_{k=1..n} T(n,k) = A350134(n).
From Mélika Tebni, Nov 24 2022: (Start)
T(n,k) = binomial(n, k)*|A069856(n-k)|.
E.g.f. column k: exp(-x)*x^k / ((1 + LambertW(-x))*k!).
T(n,k) = Sum_{j=0..n} (-1)^(j-k)*binomial(j, k)*binomial(n, j)*(n-j)^(n-j). (End)

A065456 Number of functions on n labeled nodes whose representation as a digraph has two components.

Original entry on oeis.org

0, 1, 9, 95, 1220, 18694, 334369, 6852460, 158479488, 4085349936, 116193701393, 3615197586912, 122165572502324, 4456126288810624, 174520484866919385, 7304657490838627072, 325420940777809245152, 15374940186972235659264, 767898500931828204443769

Offset: 1

John W. Layman, Nov 24 2001

nonn

			a(3)=9 since, on {1,2,3}, these functions and no others have two components: (3->1->3)(2->2), (1->3->1)(2->2), (3->2->2)(1->1), (2->3->2)(1->1), (2->1->2)(3->3), (1->2->1)(3->3), (1->2->2)(3->3), (1->3->3)(2->2) and (2->3->3)(1->1).

Alois P. Heinz, Table of n, a(n) for n = 1..386

Column k=2 of A060281.

See A001865 for the numbers of one-component (i.e. connected) functions on n labeled nodes.

katz := n->(n-1)!*sum(n^k/k!,k=0..n-1); A001865 := []; for m from 1 to 30 do A001865 := [op(A001865),katz(m)] od; A065456 := []; for n from 1 to 29 do unequal_splits := sum(binomial(n,k)*A001865[k]*A001865[n-k],k=1..floor((n-1)/2)); if (n mod 2=0) then A065456 := [op(A065456),unequal_splits+binomial(n,n/2)*(A001865[n/2])^2/2] fi; if (n mod 2=1) then A065456 := [op(A065456),unequal_splits] fi od; print(A065456); #if the connected components are of equal size, we correct the double counting. The Katz reference is at A001865. - Len Smiley, Nov 26 2001
# second Maple program:
g:= proc(n) option remember; add(n^(n-j)*(n-1)!/(n-j)!, j=1..n) end:
a:= n-> add(binomial(n, i)*g(i)*g(n-i)/2, i=0..n):
seq(a(n), n=1..20);  # Alois P. Heinz, Dec 19 2021

t=Sum[n^(n-1)x^n/n!,{n,1,20}];  Range[0, 20]! CoefficientList[Series[Log[1/(1 - t)]^2/2, {x, 0, 20}],
x] (* Geoffrey Critzer, Oct 06 2011 *)
Rest[CoefficientList[Series[Log[1+LambertW[-x]]^2, {x, 0, 20}], x]/2* Range[0, 20]!] (* Vaclav Kotesovec, Oct 05 2013 *)

x='x+O('x^20); concat([0], Vec(serlaplace(log(1+lambertw(-x))^2/2 ))) \\ G. C. Greubel, Jan 18 2018

E.g.f.: 1/2 * log(1+LambertW(-x))^2. - Vladeta Jovovic, Nov 25 2001

a(n) ~ (n-1)! * exp(n)*(log(n/2) + gamma)/4, where gamma is the Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, Oct 05 2013

More terms from Vladeta Jovovic, Nov 25 2001

A209324 Triangular array read by rows: T(n,k) is the number of endofunctions f:{1,2,...,n}-> {1,2,...,n} whose largest component has exactly k nodes; n>=1, 1<=k<=n.

Original entry on oeis.org

1, 1, 3, 1, 9, 17, 1, 45, 68, 142, 1, 165, 680, 710, 1569, 1, 855, 6290, 8520, 9414, 21576, 1, 3843, 47600, 134190, 131796, 151032, 355081, 1, 21819, 481712, 1838900, 2372328, 2416512, 2840648, 6805296, 1, 114075, 5025608, 21488292, 50609664, 48934368, 51131664, 61247664, 148869153

Offset: 1

Geoffrey Critzer, Jan 19 2013

Here component means weakly connected component in the functional digraph of f.
Row sums are n^n.
T(n,n) = A001865.
For the statistic "length of the smallest component", see A347999.

			Triangle T(n,k) begins:
  1;
  1,     3;
  1,     9,     17;
  1,    45,     68,     142;
  1,   165,    680,     710,    1569;
  1,   855,   6290,    8520,    9414,   21576;
  1,  3843,  47600,  134190,  131796,  151032,  355081;
  1, 21819, 481712, 1838900, 2372328, 2416512, 2840648, 6805296;
  ...

R. Sedgewick and P. Flajolet, Analysis of Algorithms, Addison Wesley, 1996, Chapter 8.

Alois P. Heinz, Rows n = 1..141, flattened
Steven Finch, Permute, Graph, Map, Derange, arXiv:2111.05720 [math.CO], 2021.
D. Panario and B. Richmond, Exact largest and smallest size of components, Algorithmica, 31 (2001), 413-432.

Main diagonal gives A001865.
Row sums give A000312.
Cf. A209327, A347999.

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(g(i)*
      b(n-i, max(m, i))*binomial(n-1, i-1), i=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n, 0)):
seq(T(n), n=1..12);  # Alois P. Heinz, Dec 16 2021

nn=8;t=Sum[n^(n-1)x^n/n!,{n,1,nn}];c=Log[1/(1-t)];b=Drop[Range[0,nn]!CoefficientList[Series[c,{x,0,nn}],x],1];f[list_]:=Select[list,#>0&];Map[f,Drop[Transpose[Table[Range[0,nn]!CoefficientList[Series[ Exp[Sum[b[[i]]x^i/i!,{i,1,n+1}]]-Exp[Sum[b[[i]]x^i/i!,{i,1,n}]],{x,0,nn}],x],{n,0,nn-1}]],1]]//Grid
(* Second program: *)
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[g[i]*b[n - i, Max[m, i]]* Binomial[n - 1, i - 1], {i, 1, n}]];
T[n_] := With[{p = b[n, 0]}, Table[Coefficient[p, x, i], {i, 1, n}]];
Table[T[n], {n, 1, 12}] // Flatten (* Jean-François Alcover, Dec 30 2021, after Alois P. Heinz *)

E.g.f. for column k: exp( Sum_{n=1..k} A001865(n) x^n/n!) - exp( Sum_{n=1..k-1} A001865(n) x^n/n!).

Sum_{k=1..n} k * T(n,k) = A209327(n). - Alois P. Heinz, Dec 16 2021

A350134 Number of endofunctions on [n] with at least one isolated fixed point.

Original entry on oeis.org

0, 1, 1, 10, 87, 1046, 15395, 269060, 5440463, 124902874, 3208994379, 91208536112, 2841279322871, 96258245162678, 3523457725743059, 138573785311560916, 5827414570508386335, 260928229315498155314, 12393729720071855683739, 622422708333615857463608

Offset: 0

Alois P. Heinz, Dec 15 2021

nonn

			a(3) = 10: 123, 122, 133, 132, 121, 323, 321, 113, 223, 213.

Alois P. Heinz, Table of n, a(n) for n = 0..386

Column k=1 of A347999.

Cf. A000312, A001865, A045531, A069856, A204042, A350212.

g:= proc(n) option remember; add(n^(n-j)*(n-1)!/(n-j)!, j=1..n) end:
b:= proc(n, t) option remember; `if`(n=0, t, add(g(i)*
      b(n-i, `if`(i=1, 1, t))*binomial(n-1, i-1), i=1..n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..23);

g[n_] := g[n] = Sum[n^(n - j)*(n - 1)!/(n - j)!, {j, 1, n}];
b[n_, t_] := b[n, t] = If[n == 0, t, Sum[g[i]*
     b[n - i, If[i == 1, 1, t]]*Binomial[n - 1, i - 1], {i, 1, n}]];
a[n_] := b[n, 0];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Apr 27 2022, after Alois P. Heinz *)

a(n) = A000312(n) - abs(A069856(n)).

a(n) = Sum_{k=1..n} A350212(n,k).

A350157 Total number of nodes in the smallest connected component summed over all endofunctions on [n].

Original entry on oeis.org

0, 1, 7, 61, 709, 9911, 167111, 3237921, 71850913, 1780353439, 49100614399, 1482061739423, 48873720208853, 1740252983702871, 66793644836081827, 2740470162691675711, 120029057782404141841, 5575505641199441262767, 274412698693082818767335, 14236421024010426118259883

Offset: 0

Alois P. Heinz, Dec 17 2021

nonn

			a(2) = 7 = 2 + 2 + 1 + 2: 11, 22, 12, 21.

Alois P. Heinz, Table of n, a(n) for n = 0..385

Column k=1 of A350202.

Cf. A000312, A001865, A007778, A209327, A347999.

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);

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 *)

a(n) = Sum_{k=1..n} k * A347999(n,k).

cp's OEIS Frontend

A350079 Triangle read by rows: T(n,k) is the number of endofunctions on [n] whose second-smallest component has size exactly k; n >= 0, 0 <= k <= max(0,n-1).

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A350080 Irregular triangle read by rows: T(n,k) is the number of endofunctions on [n] whose third-largest component has size exactly k; n >= 0, 0 <= k <= floor(n/3).

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A350081 Triangle read by rows: T(n,k) is the number of endofunctions on [n] whose third-smallest component has size exactly k; n >= 0, 0 <= k <= max(0,n-2).

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A350275 Irregular triangle read by rows: T(n,k) is the number of endofunctions on [n] whose fourth-largest component has size exactly k; n >= 0, 0 <= k <= floor(n/4).

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A350276 Irregular triangle read by rows: T(n,k) is the number of endofunctions on [n] whose fourth-smallest component has size exactly k; n >= 0, 0 <= k <= max(0,n-3).

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A350212 Number T(n,k) of endofunctions on [n] with exactly k isolated fixed points; triangle T(n,k), n >= 0, 0 <= k <= n, read by rows.

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A065456 Number of functions on n labeled nodes whose representation as a digraph has two components.

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