A085526 -id:A085526 - 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

A283533 a(n) = Sum_{d|n} d^(2*d + 1).

Original entry on oeis.org

1, 33, 2188, 262177, 48828126, 13060696236, 4747561509944, 2251799813947425, 1350851717672994277, 1000000000000048828158, 895430243255237372246532, 953962166440690142662256812, 1192533292512492016559195008118

Offset: 1

Seiichi Manyama, Mar 10 2017

nonn

Inverse Mobius transform of A085526. - R. J. Mathar, Mar 11 2017

			a(6) = 1^(2+1) + 2^(4+1) + 3^(6+1) + 6^(12+1) = 13060696236.

Seiichi Manyama, Table of n, a(n) for n = 1..214

Cf. Sum_{d|n} d^(k*d+1): A283498 (k=1), this sequence (k=2), A283535 (k=3).

Cf. A308696.

f[n_] := Block[{d = Divisors[n]}, Total[d^(2 d + 1)]]; Array[f, 14] (* Robert G. Wilson v, Mar 10 2017 *)

a(n) = sumdiv(n, d, d^(2*d+1)); \\ Michel Marcus, Mar 11 2017

N=20; x='x+O('x^N); Vec(x*deriv(-log(prod(k=1, N, (1-x^k)^k^(2*k))))) \\ Seiichi Manyama, Jun 18 2019

L.g.f.: -log(Product_{k>=1} (1 - x^k)^(k^(2*k))) = Sum_{k>=1} a(k)*x^k/k. - Seiichi Manyama, Jun 18 2019

A085524 a(0) = 0; a(n) = n^(2*n-1) for n > 0.

Original entry on oeis.org

0, 1, 8, 243, 16384, 1953125, 362797056, 96889010407, 35184372088832, 16677181699666569, 10000000000000000000, 7400249944258160101211, 6624737266949237011120128, 7056410014866816666030739693, 8819763977946281130444984418304, 12783403948858939111232757568359375

Offset: 0

N. J. A. Sloane, Jul 05 2003

nonn

For n > 0, a(n) is the square of the determinant of the (2*n) X (2*n) matrix with elements M(j,k) = cos(Pi*j*k/n). See the MathOverflow link. - Hugo Pfoertner, Sep 18 2021

Joerg Arndt, Table of n, a(n) for n = 0..100
Zhi-Wei Sun, Fedor Petrov, A surprising identity, discussion in MathOverflow, Jan 17 2019.

Cf. A062206, A085526, A089072.

[n eq 0 select 0 else n^(2*n-1): n in [0..30]]; // G. C. Greubel, Nov 01 2022

Join[{0}, Table[n^(2 n - 1), {n, 20}]] (* Harvey P. Dale, May 16 2016 *)

a(n) = if(n==0, 0, n^(2*n-1)) \\ Altug Alkan, Oct 04 2017

[0]+[n^(2*n-1) for n in range(1,31)] # G. C. Greubel, Nov 01 2022

a(n) = n! * [x^n] -LambertW(-n*x). - Ilya Gutkovskiy, Oct 04 2017

a(n) = A089072(2*n-1, n), n >= 1. - G. C. Greubel, Nov 01 2022

A329910 Number of harmoniously labeled graphs with n edges and at most n vertices.

Original entry on oeis.org

0, 0, 1, 4, 32, 72, 2187, 20736, 262144, 3200000, 48828125, 729000000, 13060694016, 230539333248, 4747561509943, 96717311574016, 2251799813685250, 51998697814229000, 1350851717672990000, 34867844010000000000, 1000000000000000000000, 28531167061100000000000

Offset: 1

Christian Barrientos, Nov 23 2019

A graph G with n edges is harmonious if there is an injection f from its vertex set to the group of integers modulo n such that when each edge uv of G is assigned the weight f(u)+f(v) (mod n), the resulting weights are distinct.

			a(3)=1 because there is only one harmonious graph with 3 edges and at most 3 vertices.

R. L. Graham and N. J. A. Sloane, On Additive Bases and Harmonious Graphs, SIAM J. Algebraic and Discrete Methods, 1 (1980), 382-404.
R. L. Graham and N. J. A. Sloane, On Additive Bases and Harmonious Graphs

A085526 contains the odd-indexed terms.

Table[If[EvenQ[n],(n*(n-2)/4)^(n/2),((n-1)/2)^n],{n,1,22}] (* Stefano Spezia, Nov 24 2019 *)

For n odd, a(n) = ((n-1)/2)^n. For n even, a(n) = (n*(n-2)/4)^(n/2).

cp's OEIS Frontend

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

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A283533 a(n) = Sum_{d|n} d^(2*d + 1).

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A085524 a(0) = 0; a(n) = n^(2*n-1) for n > 0.

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A329910 Number of harmoniously labeled graphs with n edges and at most n vertices.

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