A000435 -id:A000435 - 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)

A369016 Triangle read by rows: T(n, k) = binomial(n - 1, k - 1) * (k - 1)^(k - 1) * (n - k) * (n - k + 1)^(n - k - 1).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 6, 2, 0, 0, 48, 18, 12, 0, 0, 500, 192, 144, 108, 0, 0, 6480, 2500, 1920, 1620, 1280, 0, 0, 100842, 38880, 30000, 25920, 23040, 18750, 0, 0, 1835008, 705894, 544320, 472500, 430080, 393750, 326592, 0

Offset: 0

Peter Luschny, Jan 12 2024

			Triangle starts:
  [0] [0]
  [1] [0,       0]
  [2] [0,       1,      0]
  [3] [0,       6,      2,      0]
  [4] [0,      48,     18,     12,      0]
  [5] [0,     500,    192,    144,    108,      0]
  [6] [0,    6480,   2500,   1920,   1620,   1280,      0]
  [7] [0,  100842,  38880,  30000,  25920,  23040,  18750,      0]
  [8] [0, 1835008, 705894, 544320, 472500, 430080, 393750, 326592, 0]

A368849, A368982 and this sequence are alternative sum representation for A001864 with different normalizations.
T(n, k) = A368849(n, k) / n for n >= 1.
T(n, 1) = A053506(n) for n >= 1.
T(n, n - 1) = A055897(n - 1) for n >= 2.
Sum_{k=0..n} T(n, k) = A000435(n) for n >= 1.
Sum_{k=0..n} (-1)^(k+1)*T(n, k) = A368981(n) / n for n >= 1.
Cf. A066320, A369017.

T := (n, k) -> binomial(n-1, k-1)*(k-1)^(k-1)*(n-k)*(n-k+1)^(n-k-1):
seq(seq(T(n, k), k = 0..n), n=0..9);

A369016[n_, k_] := Binomial[n-1, k-1] If[k == 1, 1, (k-1)^(k-1)] (n-k) (n-k+1)^(n-k-1); Table[A369016[n, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Jan 28 2024 *)

def T(n, k): return binomial(n-1, k-1)*(k-1)^(k-1)*(n-k)*(n-k+1)^(n-k-1)
for n in range(0, 9): print([T(n, k) for k in range(n + 1)])

T = B066320 - A369017 (where B066320 = A066320 after adding a 0-column to the left and then setting offset to (0, 0)).

A036360 a(n) = Sum_{k=1..n} n! * n^(n-k+1) / (n-k)!.

Original entry on oeis.org

0, 1, 12, 153, 2272, 39225, 776736, 17398969, 435538944, 12058401393, 366021568000, 12090393761721, 431832459644928, 16585599200808937, 681703972229640192, 29858718555221585625, 1388451967046195347456, 68316647610168842824161, 3546179063131198669848576, 193670918442059606406896473

Offset: 0

N. J. A. Sloane

This formula is given as a solution to Exercise 1.15a in the Harary and Palmer reference on page 30. However, the formula may not be correct and could be a misprint for Sum_{k=2..n} n! * n^(n-k-1) / (n-k)! which is a formula for A000435(n). - Andrew Howroyd, Feb 06 2024
It appears that a(n) * n^-(n+1) is the mean position of the first duplicate in sequences of n elements randomly drawn with replacement. - Brian P Hawkins, Jan 06 2024
Total count over all mappings from [n] to [n] of tail length plus cycle size of all nodes, where mappings are sets of cycles of trees and tail length is the distance to the cycle that eventually traps the iterates of a node of the mapping; cycle size is the size of that cycle. Alternatively, number of elements on the trajectory of iterates of a node until a repeat is seen, summed over all nodes and mappings. - Marko Riedel, Jul 20 2024

			Example: Consider the map [1,2,3,4] -> [2,3,4,4]. The trajectory of node one is [1,2,3,4]. Hence the tail length is three and the cycle size is one, a fixed point.

F. Harary and E. Palmer, Graphical Enumeration, (1973), p. 30, Exercise 1.15a.
P. Flajolet and A. Odlyzko, Random Mapping Statistics, INRIA RR 1114.

N. J. A. Sloane, Table of n, a(n) for n = 0..100
Marko Riedel, Pusieux series of the tree function and random mapping statistics.

Cf. A000435, A001373, A007778, A262970.

a := proc(n) local k; add(n!*n^(n-k+1)/(n-k)!, k=0..n); end;
# Alternative, e.g.f.:
T := -LambertW(-x): egf := (T + T^2)/(1 - T)^4: ser := series(egf, x, 22):
seq(n!*coeff(ser, x, n), n = 0..19);  # Peter Luschny, Jul 20 2024

Table[Sum[n!*n^(n-k+1)/(n-k)!, {k, 1, n}], {n, 0, 19}] (* James C. McMahon, Feb 07 2024 *)
a[n_] := n E^n Gamma[n + 1, n] - n^(n + 1);
Table[a[n], {n, 0, 19}]  (* Peter Luschny, Jul 20 2024 *)

a(n) = sum(k=1, n, n! * n^(n-k+1) / (n-k)!) \\ Andrew Howroyd, Jan 06 2024

def a(n):
    total_sum = 0
    for k in range(1, n + 1):
        term = (math.factorial(n) / math.factorial(n - k))*(k**2)*(n**(n - k))
        total_sum += term
    return total_sum
# Brian P Hawkins, Jan 06 2024

a(n) = n^2 * A001865(n). - Gerald McGarvey, Apr 17 2008
a(n) = Sum_{k=1..n} n! * k^2 * n^(n-k) / (n-k)!. - Brian P Hawkins, Jan 06 2024
a(n) = n! * [z^n] (T+T^2)/(1-T)^4 where T is Cayley's tree function T(z) = Sum_{n >= 1} n^(n-1) * z^n/n!. - Marko Riedel, Jul 20 2024
a(n) ~ n^n * ((1/2) * n * sqrt(2 * Pi * n) - (1/3) * n) - Marko Riedel, Jul 20 2024
a(n) = n * e^n * Gamma(n + 1, n) - n^(n + 1) = 2*A262970(n) - A007778(n). - Peter Luschny, Jul 20 2024

Offset corrected by Brian P Hawkins, Jan 06 2024
Name edited by Andrew Howroyd, Feb 06 2024
Offset set to 0 and a(0) = 0 prepended by Marko Riedel, Jul 20 2024

A259334 Triangle read by rows: T(n,k) = k(n-1)!n^(n-k-1)/(n-k)!, 1 <= k <= n.

Original entry on oeis.org

1, 1, 1, 3, 4, 2, 16, 24, 18, 6, 125, 200, 180, 96, 24, 1296, 2160, 2160, 1440, 600, 120, 16807, 28812, 30870, 23520, 12600, 4320, 720, 262144, 458752, 516096, 430080, 268800, 120960, 35280, 5040, 4782969, 8503056, 9920232, 8817984, 6123600, 3265920, 1270080, 322560, 40320

Offset: 1

N. J. A. Sloane, Jun 25 2015

			Triangle begins:
     1;
     1,    1;
     3,    4,    2;
    16,   24,   18,    6;
   125,  200,  180,   96,   24;
  1296, 2160, 2160, 1440,  600,  120;
  ...

F. A. Haight, Overflow at a traffic light, Biometrika, 46 (1959), 420-424.
F. A. Haight, Overflow at a traffic light, Biometrika, 46 (1959), 420-424. (Annotated scanned copy)
F. A. Haight, Letter to N. J. A. Sloane, n.d.

Diagonals include A000272, A089946, A000142.

Cf. A000435.

tabl(nn) = {for (n=1, nn, for (k=1, n, print1(k*(n-1)!*n^(n-k-1)/(n-k)!, ", ");); print(););} \\ Michel Marcus, Jun 26 2015

A000435(n) = Sum_{k=0..n-1} k*T(n,k). - David desJardins, Jan 22 2017

More terms from Michel Marcus, Jun 26 2015

A320064 The number of F_2 graphs on { 1, 2, ..., n } with a unique cycle of weight 1, which corresponds to the number of reflectable bases of the root system of type D_n.

Original entry on oeis.org

0, 1, 16, 312, 7552, 220800, 7597824, 301321216, 13545271296, 681015214080, 37879390720000, 2309968030334976, 153275504883695616, 10995166075754119168, 847974316241667686400, 69971459959477921382400, 6151490510604350965940224, 574035430519008722436489216, 56669921387839814123670994944

Offset: 1

Masaya Tomie, Oct 04 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..350
Federico Ardila, Matthias Beck, and Jodi McWhirter, The Arithmetic of Coxeter Permutahedra, arXiv:2004.02952 [math.CO], 2020.
S. Azam, M. B. Soltani, M. Tomie and Y. Yoshii, A graph theoretical classification for reflectable bases, PRIMS, Vol 55 no 4, (2019), 689-736.
Theo Douvropoulos, Joel Brewster Lewis, and Alejandro H. Morales, Hurwitz numbers for reflection groups III: Uniform formulas, arXiv:2308.04751 [math.CO], 2023, see p. 11.

Cf. A000435, A001863, A038057.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( (&+[(&+[j^(j-1)*(2*x)^j/Factorial(j): j in [1..m+2]])^k/(4*k): k in [2..m+1]]) )); [0] cat [Factorial(n+1)*b[n]: n in [1..m-2]]; // G. C. Greubel, Dec 10 2018

nmax = 20; Rest[CoefficientList[Series[Sum[1/(4*m)*(Sum[k^(k-1)*(2*x)^k/k!, {k, 1, nmax}])^m, {m, 2, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!] (* Vaclav Kotesovec, Oct 23 2018 *)

seq(n)={Vec(serlaplace(sum(m=2, n, (sum(k=1, n, k^(k-1)*(2*x)^k/k!) + O(x^n))^m/(4*m))), -n)} \\ Andrew Howroyd, Nov 07 2018

apply( A320064(n)=A001863(n)*(n-1)<<(n-2), [1..20]) \\ Defines the function A320064. The additional apply(...) provides a check and illustration. - M. F. Hasler, Dec 09 2018

from math import comb
def A320064(n): return 0 if n<2 else ((sum(comb(n,k)*(n-k)**(n-k)*k**k for k in range(1,(n+1>>1)))<<1) + (0 if n&1 else comb(n,m:=n>>1)*m**n))//n<Chai Wah Wu, Apr 25-26 2023

E.g.f.: Sum_{m>=2} (1/(4*m)) (Sum_{k>=1} k^(k-1)*(2*x)^k/k!)^m.
a(n) = (n-1)*2^(n-2)*A001863(n). - M. F. Hasler, Dec 09 2018
a(n) = 2^(n-2)*A000435(n). - Chai Wah Wu, Apr 25 2023

A359628 Triangle read by rows: T(n,k) is the maximum number of connected endofunctions that are spanning subgraphs of a semi-regular loopless digraph on n vertices each with out-degree k.

Original entry on oeis.org

1, 1, 8, 1, 16, 78, 1, 32, 234, 944, 1, 64, 710, 3776, 13800, 1, 128

Offset: 2

Yali Harrary, Jan 08 2023

An endofunction represented as a digraph is one in which every vertex has out-degree 1. Loopless means that there are no fixed points in the function. The digraph of a connected endofunction is unicyclic (contains exactly one cycle).
In the case k = 1, the graphs considered have vertices with out-degree 1 and the entire graph is the only spanning subgraph that is an endofunction. Hence T(n,1) = 1.
In the case k = n-1, the graphs considered are the complete digraphs and every connected endofunction on the vertex set is a subgraph. Hence T(n, n-1) = A000435(n), which gives the total number of connected endofunctions without fixed points.

			Triangle begins:
  2 | 1;
  3 | 1,  8;
  4 | 1, 16,  78;
  5 | 1, 32, 234,  944;
  6 | 1, 64, 710, 3776, 13800;
  ...
In the following examples, the notation 1->{2,3} is shorthand for the set of arcs {(1,2), (1,3)}.
T(5,2) = 32 is attained with the digraph described by: 1->{2,3}, 2->{1,3}, 3->{1,2}, 4->{1,2}, 5->{1,2}. Regardless of the endofunction chosen, it will contain exactly one cycle and will therefore be connected, so T(5,2) = 2^5 = 32. One such endofunction is 1->2, 2->1, 3->1, 4->1, 5->1 which has the cycle between nodes 1 and 2.
T(5,3) = 234 is attained with the digraph constructed from the complete graph on 4 vertices plus a 5th vertex described by 5->{1,2,3}. The number of endofunctions which are spanning subgraphs is A000435(4)*3 = 78 * 3, since any of the 3 choices for the 5th vertex will not create a new cycle.
T(6,3) = 710 is attained with the digraph described by 1->{2,3,4}, 2->{1,3,4}, 3->{1,2,5}, 4->{3,5,6}, 5->{1,2,6}, 6->{1,2,3}. Up to isomorphism this is the only graph. Just 19 of the possible 3^6 = 729 endofunctions that are subgraphs of this digraph are disconnected. They have cycles described by one of the following permutations written in cycle notation: (13)(2456), (1456)(23), (135)(246), (146)(235), (12)(356), (13)(245), (13)(246), (145)(23), (146)(23). The last 5 of these omit one vertex which does not appear in a cycle.

Yali Harrary, Python Program for generating triangle

Cf. A000435 (connected endofunctions without fixed points).

T(n,1) = 1.
T(n,2) = 2^n.
T(n,n-1) = A000435(n).
Conjecture: T(n,n-2) = A000435(n-1)*(n-2) for n > 2.
From Andrew Howroyd, Jan 08 2023: (Start)
T(n,k) >= k*T(n-1, k) for n >= k + 2.
k^(n-k-1)*A000435(k+1) <= T(n,k) <= k^n for k < n.
(End)

A177453 Partial sums of A001863.

Original entry on oeis.org

0, 1, 5, 31, 267, 3027, 42599, 715191, 13942995, 309522515, 7707841015, 212783127799, 6449579387715, 212939326904131, 7606688596589431, 292321288041079671, 12025358303201356019, 527265684696785414387

Offset: 1

Jonathan Vos Post, May 09 2010

nonn

Partial sums of normalized total height of rooted trees with n nodes. The subsequence of primes in the partial sums begins: 5, 31, no more through a(15).

			a(5) = 0 + 1 + 4 + 26 + 236 = 267 = 3 * 89.

Cf. A000435, A001864, A001863.

A001863 := proc(n) if n = 1 then 0; else add( (n-2)!*n^k/k!,k=0..n-2) ; end if; end proc:
A177453 := proc(n) add(A001863(i),i=0..n) ; end proc: seq(A177453(n),n=1..20) ; # R. J. Mathar, May 28 2010

Accumulate[Table[Sum[(n-2)! n^k/k!,{k,0,n-2}],{n,20}]] (* Harvey P. Dale, Jun 19 2016 *)

from math import comb
def A177453(n): return sum(((sum(comb(i,k)*(i-k)**(i-k)*k**k for k in range(1,(i+1>>1)))<<1) + (0 if i&1 else comb(i,m:=i>>1)*m**i))//i//(i-1) for i in range(2,n+1)) # Chai Wah Wu, Apr 25-26 2023

a(n) = Sum_{i=1..n} A001863(i).

Extended by R. J. Mathar, May 28 2010

A321233 a(n) is the number of reflectable bases of the root system of type D_n.

Original entry on oeis.org

0, 4, 128, 4992, 241664, 14131200, 972521472, 77138231296, 6935178903552, 697359579217920, 77576992194560000, 9461629052252061696, 1255632936007234486272, 180144800985155488448512, 27786422394606966747955200, 4585649599904345055716966400, 806288164205933489807717040128

Offset: 1

Masaya Tomie, Nov 01 2018

nonn

The root systems of type D_n are only defined for n >= 4. See chapter 3 of the Humphreys reference. Sequence extended to n=1 using formula/recurrence.

J. E. Humphreys, Introduction to Lie algebras and representation theory, 2nd ed, Springer-Verlag, New York, 1972.

S. Azam, M. B. Soltani, M. Tomie and Y. Yoshii, A graph theoretical classification for reflectable bases, PRIMS, Vol 55 no 4, (2019), 689-736.

Cf. A000435, A320064.

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( (&+[ (&+[ j^(j-1)*(4*x)^j/Factorial(j) :j in [1..m+3]])^k/(4*k) :k in [2..m+2]]) )); [0] cat [Factorial(n+1)*b[n]: n in [1..m-2]]; // G. C. Greubel, Dec 09 2018

Rest[With[{m = 25}, CoefficientList[Series[Sum[Sum[j^(j - 1)*(4*x)^j/j!, {j, 1, m + 1}]^k/(4*k), {k, 2, m}], {x, 0, m}], x]*Range[0, m]!]] (* G. C. Greubel, Dec 09 2018 *)

a(n)={n!*polcoef(sum(m=2, n, (sum(k=1, n, k^(k-1)*(4*x)^k/k!) + O(x^(n-m+2)))^m/(4*m)), n)} \\ Andrew Howroyd, Nov 01 2018

A321233(n)=A001863(n)*(n-1)*4^(n-1) \\ M. F. Hasler, Dec 09 2018

from math import comb
def A321233(n): return 0 if n<2 else ((sum(comb(n,k)*(n-k)**(n-k)*k**k for k in range(1,(n+1>>1)))<<1) + (0 if n&1 else comb(n,m:=n>>1)*m**n))//n<<(n-1<<1) # Chai Wah Wu, Apr 26 2023

E.g.f.: Sum_{m>=2} (1/(4*m)) (Sum_{k>=1} k^(k-1)*(4*x)^k/k!)^m.
a(n) = 2^n*A320064(n).
a(n) = (n-1)*4^(n-1)*A001863(n). - M. F. Hasler, Dec 09 2018

A347994 a(n) = n! * Sum_{k=1..n-1} (-1)^(k+1) * n^(n-k-2) / (n-k-1)!.

Original entry on oeis.org

0, 1, 4, 30, 296, 3720, 56652, 1014832, 20909520, 487198080, 12667470740, 363607605504, 11420819358456, 389646915374080, 14349217119054300, 567315485527234560, 23967624180805666208, 1077568488585047605248, 51369752823292604784420, 2588268388538639982592000

Offset: 1

Ilya Gutkovskiy, Sep 23 2021

nonn

Cf. A000169, A000435, A133297, A134095, A347993.

Table[n! Sum[(-1)^(k + 1) n^(n - k - 2)/(n - k - 1)!, {k, 1, n - 1}], {n, 1, 20}]
nmax = 20; CoefficientList[Series[-LambertW[-x] - Log[1 - LambertW[-x]], {x, 0, nmax}], x] Range[0, nmax]! // Rest

a(n) = n! * sum(k=1, n-1, (-1)^(k+1)*n^(n-k-2)/(n-k-1)!); \\ Michel Marcus, Sep 23 2021

E.g.f.: -LambertW(-x) - log(1 - LambertW(-x)).

a(n) = A134095(n) / n.

A359686 Triangle read by rows: T(n,k) is the minimum number of connected endofunctions that are spanning subgraphs of a semi-regular loopless digraph on n vertices each with out-degree k.

Original entry on oeis.org

1, 1, 8, 0, 14, 78, 0, 22, 213, 944, 0, 0, 529, 3400, 13800, 0, 0

Offset: 2

Yali Harrary, Jan 11 2023

An endofunction represented as a digraph is one in which every vertex has out-degree 1. Loopless means that there are no fixed points in the function. The digraph of a connected endofunction is unicyclic (contains exactly one cycle).
In the case k = 1, the graphs considered have vertices with out-degree 1 and the entire graph is the only spanning subgraph that is an endofunction. Hence T(n,1) = 0. (n > 3 because when n = 2, 3 it still will be unicyclic.)
In the case k = n-1, the graphs considered are the complete digraphs and every connected endofunction on the vertex set is a subgraph. Hence T(n, n-1) = A000435(n), which gives the total number of connected endofunctions without fixed points.

			Triangle begins:
  2 | 1;
  3 | 1,  8;
  4 | 0, 14,  78;
  5 | 0, 22, 213,  944;
  6 | 0,  0, 529, 3400, 13800;
  ...
In the following examples, the notation 1->{2,3} is shorthand for the set of arcs {(1,2), (1,3)}.
T(5,2) = 22 is attained with the digraph described by: 1->{4,5}, 2->{3,5}, 3->{2,4}, 4->{1,3}, 5->{1,2}.

Cf. A000435, A359628.

T(n,1) = 0 for n > 3, otherwise 1.
T(n,n-1) = A000435(n).
T(n,k) = 0 for 2*k + 2 < n.

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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A369016 Triangle read by rows: T(n, k) = binomial(n - 1, k - 1) * (k - 1)^(k - 1) * (n - k) * (n - k + 1)^(n - k - 1).

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A036360 a(n) = Sum_{k=1..n} n! * n^(n-k+1) / (n-k)!.

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A259334 Triangle read by rows: T(n,k) = k*(n-1)!*n^(n-k-1)/(n-k)!, 1 <= k <= n.

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A320064 The number of F_2 graphs on { 1, 2, ..., n } with a unique cycle of weight 1, which corresponds to the number of reflectable bases of the root system of type D_n.

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A359628 Triangle read by rows: T(n,k) is the maximum number of connected endofunctions that are spanning subgraphs of a semi-regular loopless digraph on n vertices each with out-degree k.

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A177453 Partial sums of A001863.

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A259334 Triangle read by rows: T(n,k) = k(n-1)!n^(n-k-1)/(n-k)!, 1 <= k <= n.