A065440 -id:A065440 - cp's OEIS frontend

A350452 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.

1, 0, 0, 1, 0, 8, 0, 78, 3, 0, 944, 80, 0, 13800, 1810, 15, 0, 237432, 41664, 840, 0, 4708144, 1022252, 34300, 105, 0, 105822432, 27098784, 1286432, 10080, 0, 2660215680, 778128336, 47790540, 648900, 945, 0, 73983185000, 24165049920, 1815578160, 36048320, 138600

Offset: 0

Alois P. Heinz, Dec 31 2021

For k >= 2 and p prime, T(p,k) == 0 (mod 4*p*(p-1)). - Mélika Tebni, Jan 20 2023

			Triangle T(n,k) begins:
  1;
  0;
  0,          1;
  0,          8;
  0,         78,         3;
  0,        944,        80;
  0,      13800,      1810,       15;
  0,     237432,     41664,      840;
  0,    4708144,   1022252,    34300,    105;
  0,  105822432,  27098784,  1286432,  10080;
  0, 2660215680, 778128336, 47790540, 648900, 945;
  ...

Alois P. Heinz, Rows n = 0..200, flattened

Columns k=0-1 give: A000007, A000435.
Row sums give A065440.
T(2n,n) gives A001147.
Cf. A060281, A061356, A350446.
Cf. A071720, A007830, A106828.

c:= proc(n) option remember; add(n!*n^(n-k-1)/(n-k)!, k=2..n) end:
b:= proc(n) option remember; expand(`if`(n=0, 1, add(
      b(n-i)*binomial(n-1, i-1)*x*c(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);

c[n_] := c[n] = Sum[n!*n^(n - k - 1)/(n - k)!, {k, 2, n}];
b[n_] := b[n] = Expand[If[n == 0, 1, Sum[
     b[n - i]*Binomial[n - 1, i - 1]*x*c[i], {i, 1, n}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n/2}]][b[n]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Mar 18 2022, after Alois P. Heinz *)

\\ here AS1(n,k) gives associated Stirling numbers of 1st kind.
AS1(n,k)={(-1)^(n+k)*sum(i=0, k, (-1)^i * binomial(n, i) * stirling(n-i, k-i, 1) )}
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

From Mélika Tebni, Jan 20 2023: (Start)
E.g.f. column k: (LambertW(-x) - log(1 + LambertW(-x)))^k / k!.
-Sum_{k=1..n/2} (-1)^k*T(n,k) = A071720(n+1), for n > 0.
-Sum_{k=1..n/2} (-1)^k*T(n,k) / (n-1) = A007830(n-2), for n > 1.
T(n,k) = Sum_{j=k..n} n^(n-j)*binomial(n-1, j-1)*A106828(j, k) for n > 0. (End)

A158497 Triangle T(n,k) formed by the coordination sequences and the number of leaves for trees.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 6, 12, 1, 4, 12, 36, 108, 1, 5, 20, 80, 320, 1280, 1, 6, 30, 150, 750, 3750, 18750, 1, 7, 42, 252, 1512, 9072, 54432, 326592, 1, 8, 56, 392, 2744, 19208, 134456, 941192, 6588344, 1, 9, 72, 576, 4608, 36864, 294912, 2359296, 18874368, 150994944, 1, 10, 90, 810, 7290, 65610, 590490, 5314410, 47829690, 430467210, 3874204890

Offset: 0

Thomas Wieder, Mar 20 2009

Consider the k-fold Cartesian products CP(n,k) of the vector A(n) = [1, 2, 3, ..., n].
An element of CP(n,k) is a n-tuple T_t of the form T_t = [i_1, i_2, i_3, ..., i_k] with t=1, .., n^k.
We count members T of CP(n,k) which satisfy some condition delta(T_t), so delta(.) is an indicator function which attains values of 1 or 0 depending on whether T_t is to be counted or not; the summation sum_{CP(n,k)} delta(T_t) over all elements T_t of CP produces the count.
For the triangle here we have delta(T_t) = 0 if for any two i_j, i_(j+1) in T_t one has i_j = i_(j+1): T(n,k) = Sum_{CP(n,k)} delta(T_t) = Sum_{CP(n,k)} delta(i_j = i_(j+1)).
The test on i_j > i_(j+1) generates A158498. One gets the Pascal triangle A007318 if the indicator function tests whether for any two i_j, i_(j+1) in T_t one has i_j >= i_(j+1).
Use of other indicator functions can also calculate the Bell numbers A000110, A000045 or A000108.

			Array, A(n, k) = n*(n-1)^(k-1) for n > 1, A(n, k) = 1 otherwise, begins as:
  1,  1,   1,    1,     1,      1,       1,        1,        1, ... A000012;
  1,  1,   1,    1,     1,      1,       1,        1,        1, ... A000012;
  1,  2,   2,    2,     2,      2,       2,        2,        2, ... A040000;
  1,  3,   6,   12,    24,     48,      96,      192,      384, ... A003945;
  1,  4,  12,   36,   108,    324,     972,     2916,     8748, ... A003946;
  1,  5,  20,   80,   320,   1280,    5120,    20480,    81920, ... A003947;
  1,  6,  30,  150,   750,   3750,   18750,    93750,   468750, ... A003948;
  1,  7,  42,  252,  1512,   9072,   54432,   326592,  1959552, ... A003949;
  1,  8,  56,  392,  2744,  19208,  134456,   941192,  6588344, ... A003950;
  1,  9,  72,  576,  4608,  36864,  294912,  2359296, 18874368, ... A003951;
  1, 10,  90,  810,  7290,  65610,  590490,  5314410, 47829690, ... A003952;
  1, 11, 110, 1100, 11000, 110000, 1100000, 11000000, ............. A003953;
  1, 12, 132, 1452, 15972, 175692, 1932612, 21258732, ............. A003954;
  1, 13, 156, 1872, 22464, 269568, 3234816, 38817792, ............. A170732;
  ... ;
The triangle begins as:
  1
  1, 1;
  1, 2,  2;
  1, 3,  6,  12;
  1, 4, 12,  36,  108;
  1, 5, 20,  80,  320,  1280;
  1, 6, 30, 150,  750,  3750,  18750;
  1, 7, 42, 252, 1512,  9072,  54432, 326592;
  1, 8, 56, 392, 2744, 19208, 134456, 941192, 6588344;
  ...;
T(3,3) = 12 counts the triples (1,2,1), (1,2,3), (1,3,1), (1,3,2), (2,1,2), (2,1,3), (2,3,1), (2,3,2), (3,1,2), (3,1,3), (3,2,1), (3,2,3) out of a total of 3^3 = 27 triples in the CP(3,3).

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Array rows n: A170733 (n=14), ..., A170769 (n=50).
Columns k: A000012(n) (k=0), A000027(n) (k=1), A002378(n-1) (k=2), A011379(n-1) (k=3), A179824(n) (k=4), A101362(n-1) (k=5), 2*A168351(n-1) (k=6), 2*A168526(n-1) (k=7), 2*A168635(n-1) (k=8), 2*A168675(n-1) (k=9), 2*A170783(n-1) (k=10), 2*A170793(n-1) (k=11).
Diagonals k: A055897 (k=n), A055541 (k=n-1), A373395 (k=n-2), A379612 (k=n-3).
Sums: (-1)^n*A065440(n) (signed row).
Cf. A000045, A000108, A000110, A007318, A079901, A158498.

A158497:= func< n,k | k le 1 select n^k else n*(n-1)^(k-1) >;
[A158497(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 18 2025

A158497[n_, k_]:= If[n<2 || k==0, 1, n*(n-1)^(k-1)];
Table[A158497[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 18 2025 *)

def A158497(n,k): return n^k if k<2 else n*(n-1)^(k-1)
print(flatten([[A158497(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Mar 18 2025

T(n, k) = (n-1)^(k-1) + (n-1)^k = n*A079901(n-1,k-1), k > 0.

Sum_{k=0..n} T(n,k) = (n*(n-1)^n - 2)/(n-2), n > 2.

Edited by R. J. Mathar, Mar 31 2009

More terms added by G. C. Greubel, Mar 18 2025

A208231 Sum of the minimum cycle length over all functions f:{1,2,...,n}->{1,2,...,n} (endofunctions).

Original entry on oeis.org

0, 1, 5, 37, 373, 4761, 73601, 1336609, 27888281, 657386305, 17276807089, 500876786301, 15879053677697, 546470462226313, 20288935994319929, 808320431258439121, 34397370632215764001, 1557106493482564625793, 74713970491718324746529, 3787792171563440619543133, 202314171910557294992453009

Offset: 0

Geoffrey Critzer, Jan 10 2013

nonn

Sum of the number of endofunctions whose cycle lengths are >=i for all i >=1. A000312 + A065440 + A134362 + A208230 + ...

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

Cf. A000169, A028417, A000312, A065440, A134362, A208230, A208248, A290932.

b:= proc(n, m) option remember; `if`(n=0, m, add((j-1)!*
      b(n-j, min(m, j))*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> add(b(j$2)*n^(n-j)*binomial(n-1, j-1), j=0..n):
seq(a(n), n=0..25);  # Alois P. Heinz, May 20 2016

nn=20;t=Sum[n^(n-1)x^n/n!,{n,1,nn}];Apply[Plus,Table[Range[0,nn]!CoefficientList[Series[Exp[Sum[t^i/i,{i,n,nn}]]-1,{x,0,nn}],x],{n,1,nn}]]

E.g.f.: A(T(x)) = Sum_{k>=1} exp( Sum_{i>=k} T(x)^i/i) - 1 where A(x) is the e.g.f. for A028417 and T(x) is the e.g.f. for A000169.

A333991 a(n) = Sum_{k=0..n} (-n)^k * binomial(2n,2k).

Original entry on oeis.org

1, 0, -7, 64, -527, 3776, -7199, -712704, 28545857, -881543168, 25615822601, -733594255360, 20859188600881, -580152163418112, 15048530008948913, -311489672222081024, 713562283940993281, 511135051171610230784, -48010258775057340355559, 3439412411849176925601792

Offset: 0

Seiichi Manyama, Sep 04 2020

sign

Seiichi Manyama, Table of n, a(n) for n = 0..386

Main diagonal of A333989.

Cf. A065440, A333990.

a[0] = 1; a[n_] := Sum[(-n)^k * Binomial[2*n, 2*k], {k, 0, n}]; Array[a, 20, 0] (* Amiram Eldar, Sep 04 2020 *)
Table[Hypergeometric2F1[1/2 - n, -n, 1/2, -n], {n, 0, 20}] (* Vaclav Kotesovec, Sep 05 2020 *)
Table[Cos[2*n*ArcTan[Sqrt[n]]] * (n + 1)^n, {n, 0, 20}] // Round (* Vaclav Kotesovec, Sep 05 2020 *)

{a(n) = sum(k=0, n, (-n)^k*binomial(2*n, 2*k))}

From Vaclav Kotesovec, Sep 05 2020: (Start)
a(n) = hypergeometric2F1(1/2 - n, -n, 1/2, -n).
a(n) = (1 + i*sqrt(n))^(2*n)/2 + (1 - i*sqrt(n))^(2*n)/2, where i is the imaginary unit.
a(n) = cos(2*n*arctan(sqrt(n))) * (n + 1)^n. (End)

A350269 Triangle read by rows, T(n, k) = (n - k)!*(n - 1)^k, for 0 <= k <= n.

Original entry on oeis.org

1, 1, 0, 2, 1, 1, 6, 4, 4, 8, 24, 18, 18, 27, 81, 120, 96, 96, 128, 256, 1024, 720, 600, 600, 750, 1250, 3125, 15625, 5040, 4320, 4320, 5184, 7776, 15552, 46656, 279936, 40320, 35280, 35280, 41160, 57624, 100842, 235298, 823543, 5764801

Offset: 0

Peter Luschny, Dec 25 2021

			Triangle starts:
[0]     1
[1]     1,     0
[2]     2,     1,     1
[3]     6,     4,     4,     8
[4]    24,    18,    18,    27,    81
[5]   120,    96,    96,   128,   256,   1024
[6]   720,   600,   600,   750,  1250,   3125,  15625
[7]  5040,  4320,  4320,  5184,  7776,  15552,  46656, 279936
[8] 40320, 35280, 35280, 41160, 57624, 100842, 235298, 823543, 5764801

Cf. A000142 (first column), A001563 (second column), A000312 (subdiagonal), A065440 (main diagonal), A350268 (row sums).

A350269 := (n, k) -> (n - k)!*(n - 1)^k:
seq(seq(A350269(n, k), k = 0..n), n = 0..9);

T[n_, k_] := If[n - 1 == k == 0, 1, (n - k)! * (n - 1)^k]; Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Amiram Eldar, Dec 25 2021 *)

A350454 Number T(n,k) of endofunctions on [n] with exactly k fixed points, none of which are isolated; triangle T(n,k), n >= 0, 0 <= k <= n/2, read by rows.

Original entry on oeis.org

1, 0, 1, 2, 8, 9, 81, 76, 12, 1024, 875, 180, 15625, 12606, 2910, 120, 279936, 217217, 53550, 3780, 5764801, 4348856, 1118936, 102480, 1680, 134217728, 99111735, 26280072, 2817360, 90720, 3486784401, 2532027610, 686569050, 81864720, 3729600, 30240

Offset: 0

Alois P. Heinz, Dec 31 2021

			Triangle T(n,k) begins:
           1;
           0;
           1,          2;
           8,          9;
          81,         76,        12;
        1024,        875,       180;
       15625,      12606,      2910,      120;
      279936,     217217,     53550,     3780;
     5764801,    4348856,   1118936,   102480,    1680;
   134217728,   99111735,  26280072,  2817360,   90720;
  3486784401, 2532027610, 686569050, 81864720, 3729600, 30240;
  ...

Alois P. Heinz, Rows n = 0..200, flattened

Column k=0 gives A065440.
Row sums give |A069856|.
T(2n,n) gives A001813.
Cf. A349454.

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)+`if`(i=1, 0, 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-> exp(LambertW(-x))*(-x-LambertW(-x))^k/((1+LambertW(-x))*k!):
A350454 := (n, k)-> n! * coeff(series(egf(k), x, n+1), x, n):
seq(print(seq(A350454(n, k), k=0..n/2)), n=0..9); # Mélika Tebni, Nov 22 2022

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] + If[i == 1, 0, 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 *)

E.g.f. column k: exp(W(-x))*(-x - W(-x))^k / ((1 + W(-x))*k!), W(x) the Lambert W-function. - Mélika Tebni, Nov 22 2022
From Mélika Tebni, Dec 22 2022: (Start)
For n > 1, T(n,1) = n*A045531(n-1).
Sum_{k=0..n} (-1)^(n-k)*T(n+k,k) = 2^n.
Sum_{k=0..n} (-1)^(n-k)*T(n+k,k)/(n+k-1) = 1/n, for n > 1. (End)

A208230 Number of functions f:{1,2,...,n}->{1,2,...,n} with all cycles of length >= 4.

Original entry on oeis.org

1, 0, 0, 0, 6, 144, 3000, 64560, 1498140, 37906848, 1046608416, 31438821600, 1023129229320, 35910464987760, 1353422643322464, 54548490915316944, 2342204085734058000, 106771822456475695680, 5151207243866077428480, 262261296920723111462592, 14053817061169685865626976

Offset: 0

Geoffrey Critzer, Jan 10 2013

nonn

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

Cf. A000312, A065440, A134362.

b:= proc(n) option remember; `if`(n=0, 1, add(
      b(n-j)*binomial(n-1, j-1)*(j-1)!, j=4..n))
    end:
a:= n-> add(b(j)*n^(n-j)*binomial(n-1, j-1), j=0..n):
seq(a(n), n=0..25);  # Alois P. Heinz, May 20 2016

nn=20;t=Sum[n^(n-1)x^n/n!,{n,1,nn}];Range[0,nn]!CoefficientList[Series[Exp[Sum[t^i/i,{i,4,nn}]],{x,0,nn}],x]

E.g.f.: exp( Sum_{i>=4} T(x)^i/i ) where T(x) is the e.g.f. for A000169

A350297 Triangle read by rows: T(n,k) = n!*(n-1)^k/k!.

Original entry on oeis.org

1, 1, 0, 2, 2, 1, 6, 12, 12, 8, 24, 72, 108, 108, 81, 120, 480, 960, 1280, 1280, 1024, 720, 3600, 9000, 15000, 18750, 18750, 15625, 5040, 30240, 90720, 181440, 272160, 326592, 326592, 279936, 40320, 282240, 987840, 2304960, 4033680, 5647152, 6588344, 6588344, 5764801

Offset: 0

Robert B Fowler, Dec 23 2021

Rows n >= 2 are coefficients in a double summation power series for the integral of x^(1/x), and the integral of its inverse function y^(y^(y^(y^(...)))). See A350358.

			Triangle T(n,k) begins:
  -----------------------------------------------------------------
   n\k     0      1      2       3       4       5       6       7
  -----------------------------------------------------------------
   0  |    1,
   1  |    1,     0,
   2  |    2,     2,     1,
   3  |    6,    12,    12,      8,
   4  |   24,    72,   108,    108,     81,
   5  |  120,   480,   960,   1280,   1280,   1024,
   6  |  720,  3600,  9000,  15000,  18750,  18750,  15625,
   7  | 5040, 30240, 90720, 181440, 272160, 326592, 326592, 279936.
  ...

Cf. A000142 (first column), A062119 (second column), A065440 (main diagonal), A055897 (subdiagonal), A217701 (row sums).
Cf. A350269, A350358.
Cf. A061711, A350149.

T := (n, k) -> (n!/k!)*(n - 1)^k:
seq(seq(T(n, k), k = 0..n), n = 0..8); # Peter Luschny, Dec 24 2021

T[1, 0] := 1; T[n_, k_] := n!*(n - 1)^k/k!; Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Amiram Eldar, Dec 24 2021 *)

T(n, k) = binomial(n, k)*A350269(n, k). - Peter Luschny, Dec 25 2021

T(n+1, k) = A061711(n) * (n+1) / A350149(n, k). - Robert B Fowler, Jan 11 2022

A334014 Array read by antidiagonals: T(n,k) is the number of functions f: X->Y, where X is a subset of Y, |X| = n, |Y| = n+k, such that for every x in X, f(f(x)) != x.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 3, 2, 1, 3, 8, 18, 30, 1, 4, 15, 52, 163, 444, 1, 5, 24, 110, 478, 1950, 7360, 1, 6, 35, 198, 1083, 5706, 28821, 138690, 1, 7, 48, 322, 2110, 13482, 83824, 505876, 2954364, 1, 8, 63, 488, 3715, 27768, 203569, 1461944, 10270569, 70469000, 1, 9, 80, 702, 6078, 51894, 436656, 3618540, 29510268, 236644092, 1864204416, 1, 10, 99, 970, 9403, 90150, 854485, 8003950, 74058105, 676549450, 6098971555, 54224221050

Offset: 0

Mason C. Hart, Apr 14 2020

Comes up in the study of the Zen Stare game (see description at A134362).

T(k,n-k)*binomial(n,k)*(n-k-1)!! is the number of different possible Zen Stare rounds with n starting players and k winners.

			Array begins:
=======================================================
n\k |    0     1     2      3      4      5       6
----+--------------------------------------------------
  0 |    1     1     1      1      1      1       1 ...
  1 |    0     1     2      3      4      5       6 ...
  2 |    0     3     8     15     24     35      48 ...
  3 |    2    18    52    110    198    322     488 ...
  4 |   30   163   478   1083   2110   3715    6078 ...
  5 |  444  1950  5706  13482  27768  51894   90150 ...
  6 | 7360 28821 83824 203569 436656 854485 1557376 ...
  ...
T(2,2) = 8; This because given X = {A,B}, Y = {A,B,C,D}. The only functions f: X->Y that meet the requirement are:
f(A) = C, f(B) = C
f(A) = D, f(B) = D
f(A) = D, f(B) = C
f(A) = C, f(B) = D
f(A) = B, f(B) = C
f(A) = B, f(B) = D
f(A) = C, f(B) = A
f(A) = D, f(B) = A

Andrew Howroyd, Table of n, a(n) for n = 0..1325

Rows n=0..3 are A000012, A001477, A005563, A058794.
Columns k=0..4 are A134362, A089466, A089467, A089468, A220690(n+2).
Cf. A000169, A065440.

T(n,k)={my(w=-lambertw(-x + O(x^max(4,1+n)))); n!*polcoef(exp((k-1)*w - w^2/2)/(1-w), n)} \\ Andrew Howroyd, Apr 15 2020

T(n,k) = Sum_{i=0..n} k^(n-i)*binomial(n,i)*T(i,n-i); This means that with a constant n, T(n,k) is a polynomial of k.
T(n,0) = A134362(n).
T(0,k) = 1.
For odd n, Sum_{k=1..(n+1)/2} T(2*k-1,n-2*k+1)*binomial(n,2*k-1)*(n-2*k)!! = (n-1)^n.
E.g.f. of k-th column: exp((k-1)*W(x) - W(x)^2/2)/(1-W(x)) where W(x) is the e.g.f. of A000169. - Andrew Howroyd, Apr 15 2020

A351781 a(n) = (n-1)^n*(n-1)!^n.

Original entry on oeis.org

0, 1, 64, 104976, 8153726976, 46656000000000000, 28079296819683655680000000, 2400095991902688012207233433600000000, 37800243186554601452585666030525214621696000000000

Offset: 1

Dan Eilers, Feb 19 2022

nonn

a(n) is the number of women's ranking tables in the stable marriage problem that can be paired with a men's ranking table having no two men with the same first choice, without forming any mutual first choices. It has two terms: (n-1)^n from A065440(n), and (n-1)!^n from A091868(n-1). Such men's ranking tables having no two men with the same first choice arise in A343694, A343475, and A344663.

a(n)*A123234 is a useful alternative to A343696 which combines a Latin men's ranking table with an arbitrary women's table, since it gives fewer instances to consider.

Cf. A065440, A091868, A343694, A343475, A344663, A123234.

Mathematica
```
Table[(n-1)^n*(n-1)!^n,{n,1,9}]
```

a(n) = (n-1)^n*(n-1)!^n.

a(n) = A065440(n)*A091868(n-1).

cp's OEIS Frontend

A350452 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.

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A158497 Triangle T(n,k) formed by the coordination sequences and the number of leaves for trees.

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A208231 Sum of the minimum cycle length over all functions f:{1,2,...,n}->{1,2,...,n} (endofunctions).

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A333991 a(n) = Sum_{k=0..n} (-n)^k * binomial(2*n,2*k).

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A350269 Triangle read by rows, T(n, k) = (n - k)!*(n - 1)^k, for 0 <= k <= n.

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A350454 Number T(n,k) of endofunctions on [n] with exactly k fixed points, none of which are isolated; triangle T(n,k), n >= 0, 0 <= k <= n/2, read by rows.

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A208230 Number of functions f:{1,2,...,n}->{1,2,...,n} with all cycles of length >= 4.

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A350297 Triangle read by rows: T(n,k) = n!*(n-1)^k/k!.

A333991 a(n) = Sum_{k=0..n} (-n)^k * binomial(2n,2k).