A089072 -id:A089072 - cp's OEIS frontend

A085526 a(n) = n^(2n+1).

0, 1, 32, 2187, 262144, 48828125, 13060694016, 4747561509943, 2251799813685248, 1350851717672992089, 1000000000000000000000, 895430243255237372246531, 953962166440690129601298432, 1192533292512492016559195008117, 1728673739677471101567216945987584, 2876265888493261300027370452880859375

Offset: 0

N. J. A. Sloane, Jul 05 2003

G. C. Greubel, Table of n, a(n) for n = 0..200

Cf. A062206, A089072.

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

Table[n^(2*n+1),{n,0,20}] (* Vladimir Joseph Stephan Orlovsky, Apr 19 2011 *)

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

a(n) = n*A062206(n). - R. J. Mathar, Mar 11 2017
a(n) = [x^(2*n+1)] 1/(1 - n*x). - Ilya Gutkovskiy, Oct 10 2017
a(n) = A089072(2*n-1, n-1). - G. C. Greubel, Nov 01 2022

A352982 a(n) = Sum_{k=0..floor(n/3)} k^n.

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 65, 129, 257, 20196, 60074, 179196, 17312754, 68711380, 273234810, 31605701625, 156925970179, 780248593545, 105443761093411, 628709267031321, 3752628871164355, 580964060390826448, 4043844561787569140, 28170468954985342384

Offset: 0

Seiichi Manyama, Apr 13 2022

Seiichi Manyama, Table of n, a(n) for n = 0..458

Cf. A089072, A352945, A352981.

[(&+[k^n: k in [0..Floor(n/3)]]): n in [0..40]]; // G. C. Greubel, Nov 01 2022

a[0] = 1; a[n_] := Sum[k^n, {k, 0, Floor[n/3]}]; Array[a, 24, 0] (* Amiram Eldar, Apr 13 2022 *)

PARI
```
a(n) = sum(k=0, n\3, k^n);
```

my(N=40, x='x+O('x^N)); Vec(sum(k=0, N, (k*x)^(3*k)/(1-k*x)))

[sum( k^n for k in range((n//3)+1)) for n in range(41)] # G. C. Greubel, Nov 01 2022

G.f.: Sum_{k>=0} (k * x)^(3 * k) / (1 - k * x).

A083282 a(n) = n^(3*n).

Original entry on oeis.org

1, 64, 19683, 16777216, 30517578125, 101559956668416, 558545864083284007, 4722366482869645213696, 58149737003040059690390169, 1000000000000000000000000000000, 23225154419887808141001767796309131

Offset: 1

Christopher M. Tomaszewski (cmt1288(AT)comcast.net), Jun 02 2003

If sequence A000312 is used as the domain in the function f(n)=n^3, this sequence would be the resulting range. Also the range of the function f(n)=n^1.5 when sequence A062207 is used as the domain.

G. C. Greubel, Table of n, a(n) for n = 1..150

Cf. A000312, A062206, A062207, A089072.

[n^(3*n): n in [1..30]]; // G. C. Greubel, Nov 01 2022

Table[n^(3*n), {n, 30}] (* G. C. Greubel, Nov 01 2022 *)

[n^(3*n) for n in range(1,31)] # G. C. Greubel, Nov 01 2022

a(n) = A089072(3*n, n). - G. C. Greubel, Nov 01 2022

More terms from Michael Joseph Halm, May 16 2004

A329943 Square array read by antidiagonals: T(n,k) is the number of right total relations between set A with n elements and set B with k elements.

Original entry on oeis.org

1, 3, 1, 7, 9, 1, 15, 49, 27, 1, 31, 225, 343, 81, 1, 63, 961, 3375, 2401, 243, 1, 127, 3969, 29791, 50625, 16807, 729, 1, 255, 16129, 250047, 923521, 759375, 117649, 2187, 1, 511, 65025, 2048383, 15752961, 28629151, 11390625, 823543, 6561, 1

Offset: 1

Roy S. Freedman, Nov 24 2019

A relation R between set A with n elements and set B with k elements is a subset of the Cartesian product A x B.  A relation R is right total if for each b in B there exists an a in A such that (a,b) in R.  T(n,k) is the number of right total relations and T(k,n) is the number of left total relations: relation R is left total if for each a in A there exists a b in B such that (a,b) in R.
From Manfred Boergens, Jun 23 2024: (Start)
T(n,k) is the number of k X n binary matrices with no 0 rows.
T(n,k) is the number of coverings of [k] by tuples (A_1,...,A_n) in P([k])^n, with P(.) denoting the power set.
Swapping n,k gives A092477 (with k<=n).
For nonempty A_j see A218695 (n,k swapped).
For disjoint A_j see A089072 (n,k swapped).
For nonempty and disjoint A_j see A019538 (n,k swapped). (End)

			T(n,k) begins, for 1 <= n,k <= 9:
    1,     1,       1,         1,           1,             1,               1
    3,     9,      27,        81,         243,           729,            2187
    7,    49,     343,      2401,       16807,        117649,          823543
   15,   225,    3375,     50625,      759375,      11390625,       170859375
   31,   961,   29791,    923521,    28629151,     887503681,     27512614111
   63,  3969,  250047,  15752961,   992436543,   62523502209,   3938980639167
  127, 16129, 2048383, 260144641, 33038369407, 4195872914689, 532875860165503

Roy S. Freedman, Some New Results on Binary Relations, arXiv:1501.01914 [cs.DM], 2015.

Cf. A218695.
The diagonal T(n,n) is A055601.
A092477 = T(k,n) is the number of left total relations between A and B.
A053440 is the number of relations that are both right unique (see A329940) and right total.
A089072 is the number of functions from A to B: relations between A and B that are both right unique and left total.
A019538 is the number of surjections between A and B: relations that are right unique, right total, and left total.
A008279 is the number of injections: relations that are right unique, left total, and left unique.
A000142 is the number of bijections: relations that are right unique, left total, right total, and left unique.

Maple
```
rt:=(n,k)->(2^n-1)^k:
```

T[n_, k_] := (2^n - 1)^k; Table[T[n - k + 1, k], {n, 1, 9}, {k, 1, n}] // Flatten (* Amiram Eldar, Nov 25 2019 *)

MuPAD
```
rt:=(n,k)->(2^n-1)^k:
```

T(n,k) = (2^n - 1)^k.

A076014 Triangle in which m-th entry of n-th row is m^(n-1).

Original entry on oeis.org

1, 1, 2, 1, 4, 9, 1, 8, 27, 64, 1, 16, 81, 256, 625, 1, 32, 243, 1024, 3125, 7776, 1, 64, 729, 4096, 15625, 46656, 117649, 1, 128, 2187, 16384, 78125, 279936, 823543, 2097152, 1, 256, 6561, 65536, 390625, 1679616, 5764801, 16777216, 43046721

Offset: 1

Wolfdieter Lang, Oct 02 2002

This becomes triangle A009998(n-1, m-1), n >= m >= 1, if the m-th column entries are divided by m^(m-1).
Row sums give A076015. The m-th column (without leading zeros) gives (m^(m-1)) powers of m, m >= 1.
T(n,m) is the number of functions f:[n-1]->[(n-1)m] such that f(x)=k*x for some positive integer k <= m. Since there exactly m choices for each of the (n-1) images under f, we obtain T(n,m) = m^(n-1). - Dennis P. Walsh, Feb 27 2013
T(n+1,m+1) = (m+1)^n is the number of partial functions from an n-element set to an m-element set, n >= m >= 0. - Mohammad K. Azarian, Jun 28 2021

			For example, T(3,2)=4 since there are exactly 4 functions f from {1,2} to {1,2,3,4} that satisfy f(x)=x or f(x)=2x. If we specify each function by the ordered pair (f(1),f(2)), the four functions are (1,2), (1,4), (2,2), and (2,4). - _Dennis P. Walsh_, Feb 27 2013
Triangle begins:
  1;
  1,   2;
  1,   4,    9;
  1,   8,   27,    64;
  1,  16,   81,   256,   625;
  1,  32,  243,  1024,  3125,   7776;
  1,  64,  729,  4096, 15625,  46656, 117649;
  1, 128, 2187, 16384, 78125, 279936, 823543, 2097152;
  ...

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 2.5, p. 132.

Cf. A009998, A008279, A008277 (Stirling2).

Cf. A089072.

seq(seq(m^(n-1),m=1..n),n=1..20); # Dennis P. Walsh, Feb 27 2013

Table[m^(n-1),{n,10},{m,n}]//Flatten (* Harvey P. Dale, May 27 2017 *)

T(n, m) = m^(n-1), n >= m >= 1, otherwise 0.
G.f. for m-th column: (m^(m-1))(x^m)/(1-m*x), m >= 1.
a(n,m) = Sum_{p=1..m} Stirling2(n,p)*A008279(m-1, p-1), n >= m >= 1, otherwise 0.

A215079 Triangle T(n,k) = k^n * sum(binomial(n,n-k-j),j=0..n-k).

Original entry on oeis.org

1, 0, 1, 0, 3, 4, 0, 7, 32, 27, 0, 15, 176, 405, 256, 0, 31, 832, 3888, 6144, 3125, 0, 63, 3648, 30618, 90112, 109375, 46656, 0, 127, 15360, 216513, 1048576, 2265625, 2239488, 823543, 0, 255, 63232, 1436859, 10682368, 36328125, 62145792, 51883209, 16777216, 0, 511, 257024, 9172278, 100139008, 500000000, 1310100480, 1856265922, 1342177280, 387420489, 0, 1023, 1037312, 57159432, 889192448, 6230468750, 23339943936, 49715643824, 60129542144, 38354628411, 10000000000

Offset: 0

Olivier Gérard, Aug 02 2012

Initial term T(0,0) may be computed as 0, depending on formula and convention.

			      1
      0       1
      0       3       4
      0       7      32      27
      0      15     176     405     256
      0      31     832    3888    6144    3125
      0      63    3648   30618   90112  109375   46656
      0     127   15360  216513 1048576 2265625 2239488  823543

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Row sums sequence is A215077.
Product of A055248 and A089072 (with an initial 0 in each row).
Cf. A000225 (column k=1), A000312 (diagonal).

A215079 := proc(n,k)
    k^n*add( binomial(n,n-k-j),j=0..n-k) ;
end proc: # R. J. Mathar, Feb 08 2021

Flatten[Table[Table[Sum[k^n*Binomial[n, n - k - j], {j, 0, n - k}],  {k, 0, n}], {n, 0, 10}], 1]

T(n,k) = k^n * sum(binomial(n,n-k-j),j=0..n-k) = k^n * A055248(n,k-1).
T(n,k) = k^n * binomial(n,n-k) * 2F1(1, k-n; k+1)(-1)
T(n,1) = A000225(n). - R. J. Mathar, Feb 08 2021

A089249 Triangular array read by rows illustrating the connection between A000522 and A008292.

Original entry on oeis.org

1, 3, 4, 6, 16, 11, 10, 40, 55, 26, 15, 80, 165, 156, 57, 21, 140, 385, 546, 399, 120

Offset: 1

Alford Arnold, Dec 11 2003

The general case corresponding to other diagonals of A046802 can be derived from A046802(m,n) = Sum C(m-1,n-1) * A008292(m-r,n-1).

			The fifth row of the array is 15 80 165 156 57 resulting from A089249 (1 4 11 26 57 ) times ( 15 20 15 6 1)

Row sums = the third diagonal of A046802.

Cf. A000522, A008292, A046802, A066810, A089072.

A200146 Triangle read by rows: T(n, k) = mod(k^(n - 1), n), where 1 <= k < n.

Original entry on oeis.org

1, 1, 1, 1, 0, 3, 1, 1, 1, 1, 1, 2, 3, 4, 5, 1, 1, 1, 1, 1, 1, 1, 0, 3, 0, 5, 0, 7, 1, 4, 0, 7, 7, 0, 4, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 3, 4, 5, 0, 7, 8, 9, 4, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 2

Alonso del Arte, Nov 13 2011

Per Fermat's Little theorem, if n is prime, then row n is all 1s. However, if n is composite, that does not necessarily guarantee that the first column 1 is the only one in the row.

			The first seven rows are
1
1, 1
1, 0, 3
1, 1, 1, 1
1, 2, 3, 4, 5
1, 1, 1, 1, 1, 1
1, 0, 3, 0, 5, 0, 7
We observe that the tenth row consists of the numbers 1 to 9 in order. In base 10, the least significant digit of n^9 is the same as that of n.

Cf. A089072, A066340 (Fermat's triangle).

Column[Table[Mod[k^(n - 1), n], {n, 2, 13}, {k, n - 1}], Center] (* Nov 14 2011 *)

A344260 a(n) is the number of relations from an n-element set into a set of at most n elements.

Original entry on oeis.org

1, 3, 21, 585, 69905, 34636833, 69810262081, 567382630219905, 18519084246547628289, 2422583247133816584929793, 1268889750375080065623288448001, 2659754699919401766201267083003561985, 22306191045953951743035482794815064402563073, 748380193317489370459454048174977015562807531282433

Offset: 0

Stefano Spezia, May 13 2021

nonn

Symmetrically, also the number of relations from a set of at most n elements into an n-element set.

Row sums of A344110.

Cf. A000079, A002416, A031971, A089072, A117401, A275779.

Join[{1},Table[(2^(n+n^2)-1)/(2^n-1),{n,13}]]

a(n) = (2^(n+n^2) - 1)/(2^n - 1) for n > 0 and a(0) = 1.
a(n) ~ 2^(n^2).
a(n) = A275779(n) + 1. - Hugo Pfoertner, May 14 2021

A347034 Triangle read by columns: T(n,k) is the number of functions from an n-element set to a k-element set that are not one-to-one, k>=n>=1.

Original entry on oeis.org

0, 0, 2, 0, 3, 21, 0, 4, 40, 232, 0, 5, 65, 505, 3005, 0, 6, 96, 936, 7056, 45936, 0, 7, 133, 1561, 14287, 112609, 818503, 0, 8, 176, 2416, 26048, 241984, 2056832, 16736896, 0, 9, 225, 3537, 43929, 470961, 4601529, 42683841, 387057609, 0, 10, 280, 4960, 69760, 848800

Offset: 1

Mohammad K. Azarian, Aug 28 2021

The formula for this sequence is Theorem 2.2(iv) of the author's paper, p. 131 (see the link).

			For T(2,3): the number of functions is 3^2 and the number of one-to-one functions is 6, so 3^2 - 6 = 3 and thus T(2,3) = 3.
Triangle T(n,k) begins:
       k=1  k=2   k=3   k=4    k=5     k=6
  n=1:  0    0    0     0      0       0
  n=2:       2    3     4      5       6
  n=3:            21    40     65      96
  n=4:                  232    505     936
  n=5:                         3005    7056
  n=6:                                 45936

Mohammad K. Azarian, Remarks and Conjectures Regarding Combinatorics of Discrete Partial Functions, Int'l Math. Forum (2022) Vol. 17, No. 3, 129-141.

Cf. A000312, A002416, A036679, A068424, A089072, A101030, A199656, A344110, A344112, A344113, A344114, A344115, A344116.

A347034 := proc(n,k)
    k^n-k!/(k-n)! ;
end proc:
seq(seq(A347034(n,k),n=1..k),k=1..12) ; # R. J. Mathar, Jan 12 2023

Table[k^n - k!/(k - n)!, {k, 12}, {n, k}] // Flatten

T(n,k) = k^n - k!/(k - n)!;
row(k) = vector(k, i, T(i, k)); \\ Michel Marcus, Oct 01 2021

T(n,k) = k^n - k!/(k - n)!, k>=n.

T(n,n) = A036679(n).

cp's OEIS Frontend

A085526 a(n) = n^(2n+1).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A352982 a(n) = Sum_{k=0..floor(n/3)} k^n.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

SageMath

Formula

A083282 a(n) = n^(3*n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

Extensions

A329943 Square array read by antidiagonals: T(n,k) is the number of right total relations between set A with n elements and set B with k elements.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

MuPAD

Formula

A076014 Triangle in which m-th entry of n-th row is m^(n-1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A215079 Triangle T(n,k) = k^n * sum(binomial(n,n-k-j),j=0..n-k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A089249 Triangular array read by rows illustrating the connection between A000522 and A008292.

Views

Author

Keywords

Comments