A051129 -id:A051129 - cp's OEIS frontend

A060154 Table T(n,k) by antidiagonals of n^k mod k [n,k >= 1].

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

Offset: 1

Henry Bottomley, Mar 12 2001

			T(5,3) = 5^3 mod 3 = 125 mod 3 = 2.
Rows start:
  0, 1, 1, 1, 1, ...
  0, 0, 2, 0, 2, ...
  0, 1, 0, 1, 3, ...
  0, 0, 1, 0, 4, ...
  0, 1, 2, 1, 0, ...

Seiichi Manyama, Antidiagonals n = 1..140, flattened

Rows include A057427, A015910, A056969.
Columns include A000004, A000035 (several times), A010872, A010874, A010876, A021559 and other periodic sequences.
Diagonals include A000004 and A057427.
Cf. A114448.

T(n, k) = A051129(n, k)-n*A060155(n, k).

A060155 Table T(n,k) by antidiagonals of floor(n^k/k) [n,k >= 1].

Original entry on oeis.org

1, 0, 2, 0, 2, 3, 0, 2, 4, 4, 0, 4, 9, 8, 5, 0, 6, 20, 21, 12, 6, 0, 10, 48, 64, 41, 18, 7, 0, 18, 121, 204, 156, 72, 24, 8, 0, 32, 312, 682, 625, 324, 114, 32, 9, 0, 56, 820, 2340, 2604, 1555, 600, 170, 40, 10, 0, 102, 2187, 8192, 11160, 7776, 3361, 1024, 243, 50, 11

Offset: 1

Henry Bottomley, Mar 12 2001

			T(5,3)=[5^3/3]=[125/3]=41.
Rows start:
  1,  0,  0,   0,   0, ...
  2,  2,  2,   4,   6, ...
  3,  4,  9,  20,  48, ...
  4,  8, 21,  64, 204, ...
  5, 12, 41, 156, 625, ...

Seiichi Manyama, Antidiagonals n = 1..140, flattened

Rows include A000007, A000799, A092763, A129794, A129795, A129796, A129797, A129798, A129799, A060156.
Columns include A000027, A007590.
Diagonals include A000169.

T(n, k) = (A051129(n, k)-A060154(n, k))/k.

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).

A361291 a(n) = ((2n + 1)^n - 1)/(2n).

Original entry on oeis.org

1, 6, 57, 820, 16105, 402234, 12204241, 435984840, 17927094321, 833994048910, 43309534450633, 2483526865641276, 155867505885345241, 10627079738421409410, 782175399728156197665, 61812037545704964935440, 5220088150634922700769761, 469168161404536131943150998

Offset: 1

Stefano Spezia, Mar 12 2023

This sequence is of the form (k^n - 1)/(k - 1) with k = 2*n + 1. See crossrefs in A218722 for other sequences of the same form.

Cf. A005408, A019762 (2*e), A051129, A218722.

Cf. A000169, A000312, A038057, A052746, A062971, A213236.

Mathematica
```
Table[((2n+1)^n-1)/(2n),{n,20}]
```

def A361291(n): return (((n<<1)+1)**n-1)//(n<<1) # Chai Wah Wu, Mar 14 2023

a(n) = Sum_{i=0..n-1} A005408(n)^i.
a(n) = n! * [x^n] exp(x)*(exp(2*n*x) - 1)/(2*n).
a(n) = n! * [x^n] exp((n+1)*x)*sinh(n*x)/n.
Limit_{n->oo} a(n+1)/(n*a(n)) = 2*e.
Limit_{n->oo} (a(n+1)/a(n) - a(n)/a(n-1)) = 2*e.

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.

A347000 The (m^n)-th prime, written as square array T(n,m) read by falling antidiagonals.

Original entry on oeis.org

2, 3, 2, 5, 7, 2, 7, 23, 19, 2, 11, 53, 103, 53, 2, 13, 97, 311, 419, 131, 2, 17, 151, 691, 1619, 1543, 311, 2, 19, 227, 1321, 4637, 8161, 5519, 719, 2, 23, 311, 2309, 10627, 28687, 38873, 19289, 1619, 2, 29, 419, 3671, 21391, 79349, 171529, 180503, 65687, 3671, 2

Offset: 1

Hugo Pfoertner, Aug 10 2021

			The array begins
  2   3     5      7     11      13       17 ...
  2   7    23     53     97     151      227 ...
  2  19   103    311    691    1321     2309 ...
  2  53   419   1619   4637   10627    21391 ...
  2 131  1543   8161  28687   79349   185707 ...
  2 311  5519  38873 171529  567871  1549817 ...
  2 719 19289 180503 994837 3950183 12579617 ...

Hugo Pfoertner, Table of k, a(k) for k = 1..351, antidiagonals for m+n<=26, flattened.

Cf. A000040, A033844, A038833, A119772, A011757, A055875, A109791, A062448.

Cf. A003320, A051129.

T[n_,m_]:=Prime[m^n];Flatten[Table[Reverse[Table[T[n-m+1,m],{m,n}]],{n,10}]] (* Stefano Spezia, Aug 10 2021 *)

A354273 Square array read by ascending antidiagonals: A(n,k) = k^Omega(n).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 4, 3, 4, 1, 1, 2, 9, 4, 5, 1, 1, 4, 3, 16, 5, 6, 1, 1, 2, 9, 4, 25, 6, 7, 1, 1, 8, 3, 16, 5, 36, 7, 8, 1, 1, 4, 27, 4, 25, 6, 49, 8, 9, 1, 1, 4, 9, 64, 5, 36, 7, 64, 9, 10, 1, 1, 2, 9, 16, 125, 6, 49, 8, 81, 10, 11, 1, 1, 8, 3, 16, 25, 216, 7, 64, 9, 100, 11, 12, 1

Offset: 1

Stefano Spezia, May 22 2022

			Array begins:
    1, 1,  1,  1,   1,   1,   1,   1, ...
    1, 2,  3,  4,   5,   6,   7,   8, ...
    1, 2,  3,  4,   5,   6,   7,   8, ...
    1, 4,  9, 16,  25,  36,  49,  64, ...
    1, 2,  3,  4,   5,   6,   7,   8, ...
    1, 4,  9, 16,  25,  36,  49,  64, ...
    1, 2,  3,  4,   5,   6,   7,   8, ...
    1, 8, 27, 64, 125, 216, 343, 512, ...
    ...

K. L. Verma, On an arithmetical functions involving general exponential, Palestine Journal of Mathematics Vol. 11(2)(2022), 496-504.

Cf. A001222, A051129.

Cf. A000012 (n = 1 or k = 1), A061142 (k = 2), A165824 - A165871 (k = 3..50), A176029 (diagonal).

A[n_,k_]:=k^PrimeOmega[n]; Flatten[Table[A[n-k+1,k],{n,13},{k,n}]]

A(n, k) = A051129(A001222(n), k).

The columns are totally multiplicative: A(i*j, k) = A(i, k)*A(j, k).

cp's OEIS Frontend

A060154 Table T(n,k) by antidiagonals of n^k mod k [n,k >= 1].

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A060155 Table T(n,k) by antidiagonals of floor(n^k/k) [n,k >= 1].

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Formula

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

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PARI

Python

Formula

A361291 a(n) = ((2*n + 1)^n - 1)/(2*n).

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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.

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Maple

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A347000 The (m^n)-th prime, written as square array T(n,m) read by falling antidiagonals.

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Crossrefs

Programs

Mathematica

A354273 Square array read by ascending antidiagonals: A(n,k) = k^Omega(n).

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Programs

Mathematica

Formula

A361291 a(n) = ((2n + 1)^n - 1)/(2n).