A085524 -id:A085524 - cp's OEIS frontend

A089072 Triangle read by rows: T(n,k) = k^n, n >= 1, 1 <= k <= n.

1, 1, 4, 1, 8, 27, 1, 16, 81, 256, 1, 32, 243, 1024, 3125, 1, 64, 729, 4096, 15625, 46656, 1, 128, 2187, 16384, 78125, 279936, 823543, 1, 256, 6561, 65536, 390625, 1679616, 5764801, 16777216, 1, 512, 19683, 262144, 1953125, 10077696, 40353607, 134217728, 387420489

Offset: 1

Alford Arnold, Dec 04 2003

T(n, k) = number of mappings from an n-element set into a k-element set. - Clark Kimberling, Nov 26 2004
Let S be the semigroup of (full) transformations on [n].  Let a be in S with rank(a) = k.  Then T(n,k) = |a S|, the number of elements in the right principal ideal generated by a. - Geoffrey Critzer, Dec 30 2021
From Manfred Boergens, Jun 23 2024: (Start)
In the following two comments the restriction k<=n can be lifted, allowing all k>=1.
T(n,k) is the number of n X k binary matrices with row sums = 1.
T(n,k) is the number of coverings of [n] by tuples (A_1,...,A_k) in P([n])^k with disjoint A_j, with P(.) denoting the power set.
For nonempty A_j see A019538.
For tuples with "disjoint" dropped see A092477.
For tuples with nonempty A_j and with "disjoint" dropped see A218695. (End)

			Triangle begins:
  1;
  1,  4;
  1,  8,  27;
  1, 16,  81,  256;
  1, 32, 243, 1024,  3125;
  1, 64, 729, 4096, 15625, 46656;
  ...

Reinhard Zumkeller, Rows n = 1..100 of triangle, flattened
Mohammad K. Azarian, Remarks and Conjectures Regarding Combinatorics of Discrete Partial Functions, Int'l Math. Forum (2022) Vol. 17, No. 3, 129-141. See Theorem 2.1(ii).

Related to triangle of Eulerian numbers A008292.
Cf. A000312, A007778, A008788, A031971 (row sums), A062206, A083282.
Cf. A085524, A085526, A120485, A226065, A352981, A252982.
Cf. A019538, A092477, A218695.

a089072 = flip (^)
a089072_row n = map (a089072 n) [1..n]
a089072_tabl = map a089072_row [1..]  -- Reinhard Zumkeller, Mar 18 2013

[k^n: k in [1..n], n in [1..12]]; // G. C. Greubel, Nov 01 2022

Column[Table[k^n, {n, 8}, {k, n}], Center] (* Alonso del Arte, Nov 14 2011 *)

flatten([[k^n for k in range(1,n+1)] for n in range(1,12)]) # G. C. Greubel, Nov 01 2022

Sum_{k=1..n} T(n, k) = A031971(n).
T(n, n) = A000312(n).
T(2*n, n) = A062206(n).
a(n) = (n + T*(1-T)/2)^T, where T = round(sqrt(2*n),0). - Gerald Hillier, Apr 12 2015
T(n,k) = A051129(n,k). - R. J. Mathar, Dec 10 2015
T(n,k) = Sum_{i=0..k} Stirling2(n,i)*binomial(k,i)*i!. - Geoffrey Critzer, Dec 30 2021
From G. C. Greubel, Nov 01 2022: (Start)
T(n, n-1) = A007778(n-1), n >= 2.
T(n, n-2) = A008788(n-2), n >= 3.
T(2*n+1, n) = A085526(n).
T(2*n-1, n) = A085524(n).
T(2*n-1, n-1) = A085526(n-1), n >= 2.
T(3*n, n) = A083282(n).
Sum_{k=1..n} (-1)^k * T(n, k) = (-1)^n * A120485(n).
Sum_{k=1..floor(n/2)} T(n-k, k) = A226065(n).
Sum_{k=1..floor(n/2)} T(n, k) = A352981(n).
Sum_{k=1..floor(n/3)} T(n, k) = A352982(n). (End)

More terms and better definition from Herman Jamke (hermanjamke(AT)fastmail.fm), Jul 10 2004

Offset corrected by Reinhard Zumkeller, Mar 18 2013

A347281 a(n) = 2^(n - 1)permanent(M_n)^2 where M_n is the n X n matrix M_n(j, k) = cos(Pij*k/n).

Original entry on oeis.org

1, 2, 4, 0, 36, 288, 144, 18432, 11664, 115200, 144400, 0, 808151184, 133693952, 262440000, 299649466368, 7937314520976, 73575242956800, 21204146201616, 6459752448000000, 212406372892224, 8753824001424826368, 195844025123172289600, 152252829159294763008, 26487254903393025000000

Offset: 1

Hugo Pfoertner, Sep 18 2021

nonn

			a(7) = -3384288*cos(Pi/7) - 3460896*sin(Pi/14) - 45888*cos(2*Pi/7) - 28224*cos(15*Pi/7) + 48384*cos(17*Pi/7) + 1706400 + 3458400*sin(3*Pi/14) = 144. - _Chai Wah Wu_, Sep 19 2021

Cf. A085524, A085530, A000169, A085527, A347929.

P(n)=matpermanent(matrix(n,n,j,k,cos((Pi*j*k)/n)));
for(k=1,25,print1(round(2^(k-1)*P(k)^2),", "))

A165156 n^(2n-1)-(2n-1)^n.

Original entry on oeis.org

0, -1, 118, 13983, 1894076, 361025495, 96826261890, 35181809198207, 16677063111790072, 9999993868933742199, 7400249593980659558990, 6624737245034612579099807, 7056410013376700546645974068

Offset: 1

Vladimir Joseph Stephan Orlovsky, Sep 05 2009

sign

Cf. A085524, A085528

p=1;lst={};Do[AppendTo[lst,n^p-p^n];p=p+2,{n,4!}];lst
Table[n^(2n-1)-(2n-1)^n,{n,20}] (* Harvey P. Dale, Aug 10 2023 *)

a(n) = A085524(n)-A085528(n-1).

Definition corrected by R. J. Mathar, Sep 06 2009

A259926 a(n) = n^(2n) - n^(2n - 1).

Original entry on oeis.org

0, 8, 486, 49152, 7812500, 1813985280, 581334062442, 246290604621824, 133417453597332552, 90000000000000000000, 74002499442581601012110, 72872109936441607122321408, 84676920178401799992368876316, 114656931713301654695784797437952, 178967655284025147557258605957031250

Offset: 1

Ilya Gutkovskiy, Jul 09 2015

nonn

Cf. A005408, A005843, A062206, A085524, A086815.

[n^(2*n) - n^(2*n - 1): n in [1..20]]; // Vincenzo Librandi, Jul 10 2015

Table[n^(2 n) - n^(2 n - 1), {n, 15}]
Array[#^(2 #) - #^(2 # - 1)&, 15] (* Vincenzo Librandi, Jul 10 2015 *)

vector(20, n, n^(2*n) - n^(2*n-1)) \\ Michel Marcus, Jul 09 2015

[n**(2*n) - n**(2*n - 1) for n in range(1, 20)] # Anders Hellström, Jul 10 2015

a(n) = A062206(n) - A085524(n).
a(n) = n^A005843(n) - n^A005408(n-1).
a(n) = n! * [x^n] LambertW(-n*x)^2 / (1 + LambertW(-n*x)). - Ilya Gutkovskiy, Mar 24 2020

A355858 a(n) = n^(2n-1) mod (2n-1).

Original entry on oeis.org

0, 2, 3, 4, 8, 6, 7, 2, 9, 10, 8, 12, 18, 26, 15, 16, 29, 2, 19, 5, 21, 22, 8, 24, 18, 32, 27, 32, 50, 30, 31, 8, 63, 34, 26, 36, 37, 32, 30, 40, 80, 42, 8, 11, 45, 32, 35, 22, 49, 35, 51, 52, 8, 54, 55, 14, 57, 87, 8, 2, 94, 77, 68, 64, 113, 66, 53, 107, 69

Offset: 1

Jonas Kaiser, Jul 20 2022

nonn

If a(n) = n then 2*n-1 is prime or Fermat pseudoprime to base 2.

Cf. A000040, A001567, A006254, A085524, A174166, A179976.

a[n_] := PowerMod[n, 2*n - 1, 2*n - 1]; Array[a, 100] (* Amiram Eldar, Jul 23 2022 *)

PARI
```
a(n)=n^(2*n-1)%(2*n-1)
```

a(n)=lift(Mod(n, 2*n-1)^(2*n-1)) \\ Rémy Sigrist, Jul 21 2022

def a(n): return pow(n, 2*n-1, 2*n-1)
print([a(n) for n in range(1, 70)]) # Michael S. Branicky, Jul 23 2022

cp's OEIS Frontend

A089072 Triangle read by rows: T(n,k) = k^n, n >= 1, 1 <= k <= n.

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A347281 a(n) = 2^(n - 1)*permanent(M_n)^2 where M_n is the n X n matrix M_n(j, k) = cos(Pi*j*k/n).

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A165156 n^(2*n-1)-(2*n-1)^n.

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Mathematica

Formula

Extensions

A259926 a(n) = n^(2*n) - n^(2*n - 1).

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Magma

Mathematica

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A355858 a(n) = n^(2*n-1) mod (2*n-1).

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A347281 a(n) = 2^(n - 1)permanent(M_n)^2 where M_n is the n X n matrix M_n(j, k) = cos(Pij*k/n).

A165156 n^(2n-1)-(2n-1)^n.

A259926 a(n) = n^(2n) - n^(2n - 1).

A355858 a(n) = n^(2n-1) mod (2n-1).