cp's OEIS Frontend

Indices n such that Nexus number of 7 order (or A022523(n-1) = n^7 - (n-1)^7) is prime are listed in A121619(n) = {2,4,7,8,9,10,12,17,26, 33,36,39,41,49,51,55,59,66,78,79,80,88,96,98,...}. a(1) = 127 is Mersenne prime A000668(4).

Corresponding Nexus primes of order 7 (or primes of form A022523(n-1) = n^7 - (n-1)^7) are listed in A121618[n] = {127, 14197, 543607, 1273609, 2685817, 5217031, 16344637, 141903217,...}.

If each row started with an initial 0 (i.e., a(n,k) = (n+1)^k - n^k) then each row would be the binomial transform of the preceding row. - Henry Bottomley, May 31 2001

a(n-1, k-1) is the number of ordered k-tuples of positive integers such that the largest of these integers is n. - Alford Arnold, Sep 07 2005

From Alford Arnold, Jul 21 2006: (Start)

The sequences in A047969 can also be calculated using the Eulerian Array (A008292) and Pascal's Triangle (A007318) as illustrated below: (cf. A101095).

1 1 1 1 1 1

1 1 1 1 1 1

-----------------------------------------

1 2 3 4 5 6

1 2 3 4 5

1 3 5 7 9 11

-----------------------------------------

1 3 6 10 15 21

4 12 24 40 60

1 3 6 10

1 7 19 37 61 91

-----------------------------------------

1 4 10 20 35 56

11 44 110 220 385

11 44 110 220

1 4 10

1 15 65 175 369 671

----------------------------------------- (End)

From Peter Bala, Oct 26 2008: (Start)

The above remarks of Alford Arnold may be summarized by saying that (the transpose of) this array is the Hilbert transform of the triangle of Eulerian numbers A008292 (see A145905 for the definition of the Hilbert transform). In this context, A008292 is best viewed as the array of h-vectors of permutohedra of type A. See A108553 for the Hilbert transform of the array of h-vectors of type D permutohedra. Compare this array with A009998.

The polynomials n^k - (n-1)^k, k = 1,2,3,..., which give the nonzero entries in the columns of this array, satisfy a Riemann hypothesis: their zeros lie on the vertical line Re s = 1/2 in the complex plane. See A019538 for the connection between the polynomials n^k - (n-1)^k and the Stirling polynomials of the simplicial complexes dual to the type A permutohedra.

(End)

Empirical: (n+1)^(k+1) - n^(k+1) is the number of first differences of length k+1 arrays of numbers in 0..n, k > 0. - R. H. Hardin, Jun 30 2013

a(n-1, k-1) is the number of bargraphs of width k and height n. Examples: a(1,2) = 7 because we have [1,1,2], [1,2,1], [2,1,1], [1,2,2], [2,1,2], [2,2,1], and [2,2,2]; a(2,1) = 5 because we have [1,3], [2,3], [3,1], [3,2], and [3,3] (bargraphs are given as compositions). This comment is equivalent to A. Arnold's Sep 2005 comment. - Emeric Deutsch, Jan 30 2017

Original name: The first summation of row 4 of Euler's triangle - a row that will recursively accumulate to the power of 4.

All Mersenne primes of form 2^p-1 = {3, 7, 31, 127, 8191,...} belong to a(n). Mersenne prime A000668(n) = a(k) when prime(k) = A000043(n). Last digit is always 1 for Nexus numbers of form n^p - (n-1)^p with p = {5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101,...} = A004144(n) Pythagorean primes: primes of form 4n+1.

Original Name: Shells (nexus numbers) of shells of shells of shells of the power of 5.

The (Worpitzky/Euler/Pascal Cube) "MagicNKZ" algorithm is: MagicNKZ(n,k,z) = Sum_{j=0..k+1} (-1)^j*binomial(n + 1 - z, j)*(k - j + 1)^n, with k>=0, n>=1, z>=0. MagicNKZ is used to generate the n-th accumulation sequence of the z-th row of the Euler Triangle (A008292). For example, MagicNKZ(3,k,0) is the 3rd row of the Euler Triangle (followed by zeros) and MagicNKZ(10,k,1) is the partial sums of the 10th row of the Euler Triangle. This sequence is MagicNKZ(5,k-1,2).

Row sums = A000312.

Main diagonal = A045531.