A051128 -id:A051128 - cp's OEIS frontend

A003992 Square array read by upwards antidiagonals: T(n,k) = n^k for n >= 0, k >= 0.

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 4, 1, 0, 1, 4, 9, 8, 1, 0, 1, 5, 16, 27, 16, 1, 0, 1, 6, 25, 64, 81, 32, 1, 0, 1, 7, 36, 125, 256, 243, 64, 1, 0, 1, 8, 49, 216, 625, 1024, 729, 128, 1, 0, 1, 9, 64, 343, 1296, 3125, 4096, 2187, 256, 1, 0, 1, 10, 81, 512, 2401, 7776, 15625, 16384, 6561, 512, 1, 0

Offset: 0

Marc LeBrun

If the array is transposed, T(n,k) is the number of oriented rows of n colors using up to k different colors. The formula would be T(n,k) = [n==0] + [n>0]*k^n. The generating function for column k would be 1/(1-k*x). For T(3,2)=8, the rows are AAA, AAB, ABA, ABB, BAA, BAB, BBA, and BBB. - Robert A. Russell, Nov 08 2018

T(n,k) is the number of multichains of length n from {} to [k] in the Boolean lattice B_k. - Geoffrey Critzer, Apr 03 2020

			Rows begin:
[1, 0,  0,   0,    0,     0,      0,      0, ...],
[1, 1,  1,   1,    1,     1,      1,      1, ...],
[1, 2,  4,   8,   16,    32,     64,    128, ...],
[1, 3,  9,  27,   81,   243,    729,   2187, ...],
[1, 4, 16,  64,  256,  1024,   4096,  16384, ...],
[1, 5, 25, 125,  625,  3125,  15625,  78125, ...],
[1, 6, 36, 216, 1296,  7776,  46656, 279936, ...],
[1, 7, 49, 343, 2401, 16807, 117649, 823543, ...], ...

T. D. Noe, Rows n = 0..50 of triangle, flattened

Rows 0-49 are A000007, A000012, A000079, A000244, A000302, A000351, A000400, A000420, A001018, A001019, A011557, A001020, A001021, A001022, A001023, A001024, A001025, A001026, A001027, A001029, A009964-A009992, A087752.
Columns 0-26 are A000012, A001477, A000290, A000578, A000583, A000584, A001014, A001015, A001016, A001017, A008454, A008455, A008456, A010801-A010813, A089081.
Main diagonal is A000312. Other diagonals include A000169, A007778, A000272, A008788. Antidiagonal sums are in A026898.
Cf. A099555.
Transpose is A004248. See A051128, A095884, A009999 for other versions.
Cf. A277504 (unoriented), A293500 (chiral).

[[(n-k)^k: k in [0..n]]: n in [0..10]]; // G. C. Greubel, Nov 08 2018

Table[If[k == 0, 1, (n - k)^k], {n, 0, 11}, {k, 0, n}]//Flatten

T(n,k) = (n-k)^k \\ Charles R Greathouse IV, Feb 07 2017

E.g.f.: Sum T(n,k)*x^n*y^k/k! = 1/(1-x*exp(y)). - Paul D. Hanna, Oct 22 2004

E.g.f.: Sum T(n,k)*x^n/n!*y^k/k! = e^(x*e^y). - Franklin T. Adams-Watters, Jun 23 2006

More terms from David W. Wilson

Edited by Paul D. Hanna, Oct 22 2004

A051129 Table T(n,k) = k^n read by upwards antidiagonals (n >= 1, k >= 1).

Original entry on oeis.org

1, 1, 2, 1, 4, 3, 1, 8, 9, 4, 1, 16, 27, 16, 5, 1, 32, 81, 64, 25, 6, 1, 64, 243, 256, 125, 36, 7, 1, 128, 729, 1024, 625, 216, 49, 8, 1, 256, 2187, 4096, 3125, 1296, 343, 64, 9, 1, 512, 6561, 16384, 15625, 7776, 2401, 512, 81, 10, 1, 1024, 19683, 65536, 78125, 46656, 16807, 4096, 729, 100, 11

Offset: 1

N. J. A. Sloane

(n-th term) = (n-th term of A002260)^(n-th term of A004736). Both A002260 and A004736 are related to A002024. - Robert A. Stump (bee_ess107(AT)yahoo.com), Aug 29 2002

			  1   2       3       4       5       6       7
  1   4       9      16      25      36      49
  1   8      27      64     125     216     343
  1  16      81     256     625    1296    2401
  1  32     243    1024    3125    7776   16807
  1  64     729    4096   15625   46656  117649
  1 128    2187   16384   78125  279936  823543

T. D. Noe, Rows n = 1..50 of triangle, flattened

Cf. A051128 (transposed), A003992 (transposed), A004248.
Cf. A002024, A002260, A004736.
Cf. A002260, A003101 (antidiagonal sums), A000169 (central terms), A003320 (row maxima), A247358 (sorted rows).

a051129 n k = k ^ (n - k)
a051129_row n = a051129_tabl !! (n-1)
a051129_tabl = zipWith (zipWith (^)) a002260_tabl $ map reverse a002260_tabl
-- Reinhard Zumkeller, Sep 14 2014

T:= (n, k)-> k^n:
seq(seq(T(1+d-k, k), k=1..d), d=1..11);  # Alois P. Heinz, Apr 18 2020

Table[ k^(n-k+1), {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 30 2012 *)

b(n) = floor(1/2 + sqrt(2 * n));
vector(100, n, (n - b(n) * (b(n) - 1) / 2)^(b(n) * (b(n) + 1) / 2 - n + 1)) \\ Altug Alkan, Dec 09 2015

a(n) = (n - b(n) * (b(n) - 1) / 2)^(b(n) * (b(n) + 1) / 2 - n + 1), where b(n) = [ 1/2 + sqrt(2 * n) ]. (b(n) is the n-th term of A002024.) - Robert A. Stump (bee_ess107(AT)yahoo.com), Aug 29 2002

More terms from James Sellers, Dec 11 1999

A367580 Multiset multiplicity kernel (MMK) of n. Product of (least prime factor with exponent k)^(number of prime factors with exponent k) over all distinct exponents k appearing in the prime factorization of n.

Original entry on oeis.org

1, 2, 3, 2, 5, 4, 7, 2, 3, 4, 11, 6, 13, 4, 9, 2, 17, 6, 19, 10, 9, 4, 23, 6, 5, 4, 3, 14, 29, 8, 31, 2, 9, 4, 25, 4, 37, 4, 9, 10, 41, 8, 43, 22, 15, 4, 47, 6, 7, 10, 9, 26, 53, 6, 25, 14, 9, 4, 59, 18, 61, 4, 21, 2, 25, 8, 67, 34, 9, 8, 71, 6, 73, 4, 15, 38

Offset: 1

Gus Wiseman, Nov 26 2023

nonn

As an operation on multisets, this is represented by A367579.

			90 has prime factorization 2^1 * 3^2 * 5^1, so for k = 1 we have 2^2, and for k = 2 we have 3^1, so a(90) = 12.

Positions of 2's are A000079 without 1.
Positions of 3's are A000244 without 1.
Positions of primes (including 1) are A000961.
Positions of prime(k) are prime powers prime(k)^i, rows of A051128.
Depends only on rootless base A052410, see A007916.
Positions of prime powers are A072774.
Positions of squarefree numbers are A130091.
Agrees with A181819 at positions A367683, counted by A367682.
Rows of A367579 have this rank, sum A367581, max A367583, min A055396.
Positions of first appearances are A367584, sorted A367585.
Positions of powers of 2 are A367586.
Divides n at positions A367685, counted by A367684.
The opposite version (cokernel) is A367859.
A007947 gives squarefree kernel.
A027746 lists prime factors, length A001222, indices A112798.
A027748 lists distinct prime factors, length A001221, indices A304038.
A071625 counts distinct prime exponents.
A124010 gives multiset of multiplicities (prime signature), sorted A118914.
Cf. A001597, A005117, A020639, A051904, A052409, A062770, A175781, A238747, A288636, A353742, A367582.

mmk[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Min@@Select[q,Count[q,#]==i&], {i,mts}]]];
Table[Times@@mmk[Join@@ConstantArray@@@FactorInteger[n]], {n,100}]

a(n^k) = a(n) for all positive integers n and k.
A001221(a(n)) = A071625(n).
A001222(a(n)) = A001221(n).
If n is squarefree, a(n) = A020639(n)^A001222(n).
A056239(a(n)) = A367581(n).

A009999 Triangle in which j-th entry in i-th row is (i+1-j)^j, 0<=j<=i.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 9, 8, 1, 1, 5, 16, 27, 16, 1, 1, 6, 25, 64, 81, 32, 1, 1, 7, 36, 125, 256, 243, 64, 1, 1, 8, 49, 216, 625, 1024, 729, 128, 1, 1, 9, 64, 343, 1296, 3125, 4096, 2187, 256, 1, 1, 10, 81, 512, 2401, 7776, 15625, 16384, 6561, 512, 1

Offset: 0

N. J. A. Sloane

T(n,k) is the number of ways of placing 1..k in n boxes such that each box contains at most one number, and numbers in adjacent boxes are in increasing order. This can be proved by observing that there are n-(k-1) ways of extending each of T(n-1,k-1). - Jimin Park, Apr 16 2023

The n-th diagonal consists of n^k. This can also be generated as the Akiyama-Tanigawa algorithm applied to the sequence binomial(n+k,k). - Shel Kaphan, May 03 2024

			Triangle begins
  1
  1  1
  1  2  1
  1  3  4   1
  1  4  9   8    1
  1  5 16  27   16    1
  1  6 25  64   81   32     1
  1  7 36 125  256  243    64     1
  1  8 49 216  625 1024   729   128    1
  1  9 64 343 1296 3125  4096  2187  256   1
  1 10 81 512 2401 7776 15625 16384 6561 512 1

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 24.

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Row sums give A026898.
T(2n,n) gives A000169(n+1).
Cf. A095884, A051128.
Cf. A009998 (mirrored).

a009999 n k = (n + 1 - k) ^ k
a009999_row n = a009999_tabl !! n
a009999_tabl = [1] : map snd (iterate f ([1,1], [1,1])) where
   f (us@(u:_), vs) = (us', 1 : zipWith (*) us' vs)
                      where us' = (u + 1) : us
-- Reinhard Zumkeller, Feb 02 2014

A009999 := proc(i,j) (i+1-j)^j ; end proc: # R. J. Mathar, Jan 16 2011

Table[(i+1-j)^j, {i, 0, 10}, {j, 0, i}] // Flatten (* Jean-François Alcover, Jan 14 2014 *)

T(n,0) = 1; T(n,k) = (n-k+1)*T(n-1,k-1) for k=1..n. - Reinhard Zumkeller, Feb 02 2014

T(n,k) = Sum_{i=0..k} binomial(k,i)*T(n-1-i,k-i). - Jimin Park, Apr 16 2023

T(10,8) corrected by Reinhard Zumkeller, Feb 02 2014

A332101 Least m such that m^n <= Sum_{k

Original entry on oeis.org

2, 3, 5, 6, 8, 9, 11, 12, 14, 15, 16, 18, 19, 21, 22, 24, 25, 27, 28, 29, 31, 32, 34, 35, 37, 38, 40, 41, 42, 44, 45, 47, 48, 50, 51, 52, 54, 55, 57, 58, 60, 61, 63, 64, 65, 67, 68, 70, 71, 73, 74, 76, 77, 78, 80, 81, 83, 84, 86, 87, 89, 90, 91, 93, 94, 96, 97

Offset: 0

M. F. Hasler, Apr 14 2020

nonn

In a list (1^n, 2^n, 3^n, ...) (rows of table A051128 or A051129), a(n) is the index of the first term less than or equal to the sum of all earlier terms, cf. example.

Obviously a lower bound for any s solution to s^n = Sum_{x in S} x^n, S subset of {1, ..., s-1}, cf. A030052.

			For n = 0, m^0 > Sum_{0 < k < m} k^0 = 0 for m = 0, 1 (empty sums), but 2^0 = Sum_{0 < k < 2} k^0 = 1, so a(0) = 2.
For n = 1, 1^1 > Sum_{0 < k < 1} k^1 = 0 (empty sum) and 2^1 > Sum_{0 < k < 2} k^1 = 1, but 3^1 <= Sum_{0 < k < 3} k^1 = 1 + 2, so a(1) = 3.
To find a(n) one can add up terms in row n of the table k^n until the sum equals or exceeds the next term, whose column number k is then a(n):
  n |k: 1  2   3   4    5    6          Comment
  --+---------------------------------------------------------------
  1 |  1   2   3                  1 < 2 but 1 + 2 >= 3, so a(1) = 3.
  2 |  1   4   9  16   25         1 + 4 + 9 + 16 > 25, and a(2) = 5.
  3 |  1   8  27  64  125  216    1 + 8 + 27 + 64 + 125 > 216: a(3) = 6.

Michael De Vlieger, Table of n, a(n) for n = 0..1000
Brennan Benfield and Oliver Lippard, Integers that are not the sum of positive powers, arXiv:2404.08193 [math.NT], 2024.

Cf. A051128, A051129.

Cf. A078607, A332097 (maximum of E(s), cf comments), A030052 (least k such that k^n = sum of distinct n-th powers), A332065 (all k such that k^n is a sum of distinct n-th powers).

Table[Block[{m = 1, s = 0}, While[m^n > s, s = s + m^n; m++]; m], {n, 0, 66}] (* Michael De Vlieger, Apr 30 2020 *)

PARI
```
apply( A332101(n,s)=for(m=1,oo, s
				
```

a(n) = round(n / log(2)) + 2. (Conjectured; verified up to 10^4, in particular for 3525/log(2) = 5085.500019... and 7844/log(2) ~ 11316.49990...)

a(n) = A078607(n) + 2 for almost all n > 1. (n = 777451915729368 might be an exception to this equality or the above one.) - M. F. Hasler, May 08 2020

A095884 Triangle read by rows: T(n,k) = (n-k)^k, n>=1, 1<=k<=n.

Original entry on oeis.org

0, 1, 0, 2, 1, 0, 3, 4, 1, 0, 4, 9, 8, 1, 0, 5, 16, 27, 16, 1, 0, 6, 25, 64, 81, 32, 1, 0, 7, 36, 125, 256, 243, 64, 1, 0, 8, 49, 216, 625, 1024, 729, 128, 1, 0, 9, 64, 343, 1296, 3125, 4096, 2187, 256, 1, 0, 10, 81, 512, 2401, 7776, 15625, 16384, 6561, 512, 1, 0, 11, 100

Offset: 1

Herman Jamke (hermanjamke(AT)fastmail.fm), Jul 10 2004

			0
1 0
2 1 0
3 4 1 0
4 9 8 1 0
5 16 27 16 1 0
6 25 64 81 32 1 0
7 36 125 256 243 64 1 0
8 49 216 625 1024 729 128 1 0
9 64 343 1296 3125 4096 2187 256 1 0

Cf. A051128, A003992, A009999.

A220417 Table T(n,k) = k^n - n^k, n, k > 0, read by descending antidiagonals.

Original entry on oeis.org

0, 1, -1, 2, 0, -2, 3, 1, -1, -3, 4, 0, 0, 0, -4, 5, -7, -17, 17, 7, -5, 6, -28, -118, 0, 118, 28, -6, 7, -79, -513, -399, 399, 513, 79, -7, 8, -192, -1844, -2800, 0, 2800, 1844, 192, -8, 9, -431, -6049, -13983, -7849, 7849, 13983, 6049, 431, -9, 10, -924, -18954, -61440, -61318, 0, 61318, 61440, 18954, 924, -10

Offset: 1

Boris Putievskiy, Dec 14 2012

			The table T(n,k) (with rows n >= 1 and columns k >= 1) begins as follows:
   0   1     2      3       4       5 ...
  -1   0     1      0      -7     -28 ...
  -2  -1     0    -17    -118    -513 ...
  -3   0    17      0    -399   -2800 ...
  -4   7   118    399       0   -7849 ...
  -5  28   513   2800    7849       0 ...
  ...
The start of the sequence as a triangular array, read by rows (i.e., descending antidiagonals of T(n,k)), is as follows:
  0;
  1,  -1;
  2,   0,   -2;
  3,   1,   -1, -3;
  4,   0,    0,  0,  -4;
  5,  -7,  -17, 17,   7, -5;
  6, -28, -118,  0, 118, 28, -6;
  ...
In the above triangle, row number m contains m numbers: m^1 - 1^m, (m-1)^2 - 2^(m-1), ..., 1^m - m^1.

Boris Putievskiy, Rows n = 1..76 of triangle, flattened

Cf. A002260, A004736, A051128, A051129, A055651, A156353.

matrix(9, 9, n, k, k^n - n^k) \\ Michel Marcus, Oct 04 2019

t=int((math.sqrt(8*n-7) - 1)/ 2)
m=((t*t+3*t+4)/2-n)**(n-t*(t+1)/2)-(n-t*(t+1)/2)**((t*t+3*t+4)/2-n)

As a linear array, the sequence is a(n) = A004736(n)^A002260(n) - A002260(n)^A004736(n) or

a(n) = ((t*t + 3*t + 4)/2 - n)^(n - t*(t + 1)/2) - (n - t*(t + 1)/2)^((t*t + 3*t + 4)/2 - n) where t = floor((-1 + sqrt(8*n - 7))/2).

A352001 Square array A(n, k), n, k >= 1, read by antidiagonals upwards; A(n, k) = Product_{ i >= 1 } prime(k*i)^e_i where n = Product_{ i >= 1 } prime(i)^e_i (where prime(i) denotes the i-th prime number).

Original entry on oeis.org

1, 2, 1, 3, 3, 1, 4, 7, 5, 1, 5, 9, 13, 7, 1, 6, 13, 25, 19, 11, 1, 7, 21, 23, 49, 29, 13, 1, 8, 19, 65, 37, 121, 37, 17, 1, 9, 27, 37, 133, 47, 169, 43, 19, 1, 10, 49, 125, 53, 319, 61, 289, 53, 23, 1, 11, 39, 169, 343, 71, 481, 73, 361, 61, 29, 1

Offset: 1

Rémy Sigrist, Feb 27 2022

In other words, in prime factorization of n, replace prime(i) by prime(k*i).

For any k >= 1, n -> A(n, k) is completely multiplicative.

			Square array A(n, k) begins:
  n\k|   1   2    3    4     5     6     7     8      9     10
  ------------------------------------------------------------
    1|   1   1    1    1     1     1     1     1      1      1
    2|   2   3    5    7    11    13    17    19     23     29
    3|   3   7   13   19    29    37    43    53     61     71
    4|   4   9   25   49   121   169   289   361    529    841
    5|   5  13   23   37    47    61    73    89    103    113
    6|   6  21   65  133   319   481   731  1007   1403   2059
    7|   7  19   37   53    71    89   107   131    151    173
    8|   8  27  125  343  1331  2197  4913  6859  12167  24389
    9|   9  49  169  361   841  1369  1849  2809   3721   5041
   10|  10  39  115  259   517   793  1241  1691   2369   3277

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (antidiagonals n = 1..150 flattened)

Cf. A000040, A001248, A031215, A051128, A297002, A351995.

Main diagonal gives A352028.

A:= (n, k)-> mul(ithprime(k*numtheory[pi](i[1]))^i[2], i=ifactors(n)[2]):
seq(seq(A(d+1-k, k), k=1..d), d=1..12);  # Alois P. Heinz, Feb 28 2022

Table[If[# == 1, 1, Times @@ Map[Prime[#3*PrimePi[#1]]^#2 & @@ Flatten[{#1, k}] &, FactorInteger[#]]] &[n - k + 1], {n, 11}, {k, n}] // Flatten (* Michael De Vlieger, Feb 28 2022 *)

A(n,k) = { my (f=factor(n)); prod (i=1, #f~, prime(k * primepi(f[i,1])) ^ f[i,2]) }

A(A(n, k), k') = A(n, k*k').
A(n, 1) = n.
A(n, 2) = A297002(n).
A(1, k) = 1.
A(2, k) = prime(k) (the k-th prime number).
A(3, k) = prime(2*k) = A031215(k).
A(4, k) = A001248(k).

A114448 Array a(n,k) = n^k (mod k) read by antidiagonals (k>=1, n>=1).

Original entry on oeis.org

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

Offset: 1

Leroy Quet, Feb 14 2006

Alternate description: triangular array a(n, k) = n^k (mod k) read by rows (n > 1, 0 < k < n). This is equivalent because a(n, k) = a(n-k, k). - David Wasserman, Jan 25 2007

			2^6 = 64 and 64 (mod 6) is 4. So a(2,6) = 4.

Cf. A051127, A051128, A094263. Transpose of A060154. First 13 rows are A057427(n-1), A015910, A066601, A066602, A066603, A066604, A066438, A066439, A066440, A056969, A066441, A066442, A116609. First 12 columns are A000004, A000035, A010872, A000035, A010874, A070431, A010876, A000035, A021559(n+1), A008959, A010880, A070435.

a[n_, k_] := Mod[n^k, k]; Table[a[n - k + 1, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Dec 12 2012 *)

More terms from David Wasserman, Jan 25 2007

A329940 Square array read by antidiagonals upwards: T(n,k) is the number of right unique relations between set A with n elements and set B with k elements.

Original entry on oeis.org

1, 3, 2, 7, 8, 3, 15, 26, 15, 4, 31, 80, 63, 24, 5, 63, 242, 255, 124, 35, 6, 127, 728, 1023, 624, 215, 48, 7, 255, 2186, 4095, 3124, 1295, 342, 63, 8, 511, 6560, 16383, 15624, 7775, 2400, 511, 80, 9, 1023, 19682, 65535, 78124, 46655, 16806, 4095, 728, 99, 10

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 unique if (a, b1) in R and (a,b2) in R implies b1=b2. T(n,k) is the number of right unique relations and T(k,n) is the number of left unique relations: relation R is left unique if (a1,b) in R and (a2,b) in R implies a1=a2.

			T(n,k) begins:
    1,    2,     3,      4,       5,       6,        7,        8, ...
    3,    8,    15,     24,      35,      48,       63,       80, ...
    7,   26,    63,    124,     215,     342,      511,      728, ...
   15,   80,   255,    624,    1295,    2400,     4095,     6560, ...
   31,  242,  1023,   3124,    7775,   16806,    32767,    59048, ...
   63,  728,  4095,  15624,   46655,  117648,   262143,   531440, ...
  127, 2186, 16383,  78124,  279935,  823542,  2097151,  4782968, ...
  255, 6560, 65535, 390624, 1679615, 5764800, 16777215, 43046720, ...

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

Cf. A037205 (main diagonal).

Cf. A003992, A004248, A009998, A009999, A051128, A051129, A095884.

T:= (n, k)-> (k+1)^n-1:
seq(seq(T(1+d-k, k), k=1..d), d=1..12);

T[n_, k_] := (k + 1)^n - 1; Table[T[n - k + 1, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Amiram Eldar, Nov 25 2019 *)

MuPAD
```
T:=(n,k)->(k+1)^n-1:
```

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

cp's OEIS Frontend

A003992 Square array read by upwards antidiagonals: T(n,k) = n^k for n >= 0, k >= 0.

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A051129 Table T(n,k) = k^n read by upwards antidiagonals (n >= 1, k >= 1).

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A367580 Multiset multiplicity kernel (MMK) of n. Product of (least prime factor with exponent k)^(number of prime factors with exponent k) over all distinct exponents k appearing in the prime factorization of n.

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A009999 Triangle in which j-th entry in i-th row is (i+1-j)^j, 0<=j<=i.

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A332101 Least m such that m^n <= Sum_{k

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A095884 Triangle read by rows: T(n,k) = (n-k)^k, n>=1, 1<=k<=n.

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A220417 Table T(n,k) = k^n - n^k, n, k > 0, read by descending antidiagonals.

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