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The coefficients in the expansion of 1/(1 - x - 2*x^5) are given by the sequence generated by the row sums in triangle A318775.

Coefficients in expansion of 1/(1 - x - 2*x^5) are given by the sum of numbers along "fourth Layer" skew diagonals pointing top-right in triangle A013609 ((1+2x)^n) and by the sum of numbers along "fourth Layer" skew diagonals pointing top-left in triangle A038207 ((2+x)^n), see links.

To be included, a polyhypercube should have all the 2^n*n! symmetries of the n-dimensional hypercube.

Let n >= 1 and m = A171977(n). Then T(n,k) = 0 if neither k nor k-1 is a multiple of m. Also, there exists a number K such that T(n,k) > 0 if k >= K and either k or k-1 is a multiple of m. In particular, T(n,k) > 0 for all sufficiently large k if and only if n is odd. Sketch of proof: Assume that the point of rotation of the symmetries is in the origin and that the center of each cell have integer or half-integer coordinates, depending on whether the point of rotation is in the center of a cell or at the common corner of 2^n cells. The number of cells that are equivalent to a given cell c is n!/(x_0!*x_1!*...)*2^(n-x_0), where x_1, x_2, ... are the frequencies of the absolute values of the nonzero coordinates of c and x_0 is the number of zero coordinates of c. It can be proved that this number is divisible by m unless c is the cell at the origin (in which case x_0 = n and there are no other equivalent cells). (It is sufficient to check the case where all nonzero coordinates have the same absolute value, i.e., that all numbers except 1 in the n-th row of A013609 are divisible by m; the other numbers are multiples of these.) Since either none or all of the cells equivalent to a given cell must be part of the polyhypercube, this proves the first part. For the second part, say that a cell where the absolute values of all coordinates are equal and nonzero is a corner cell, and that a cell with a single nonzero coordinate is a spike cell. Corner cells and spike cells come in sets of 2^n and 2*n equivalent cells, respectively, and the GCD of 2^n and 2*n is already equal to m. Assume that n >= 3 (the case n <= 2 is easily handled), that k >= (4*n-1)^n, and that either k or k-1 is a multiple of m. Start with a solid cube made up of (4*n-1)^n cells. Remove the central cell if k is even, so that the number of remaining cells is congruent to k (mod m). Since GCD(2^n,2*n) = m, we can remove at most 2*n-1 sets of 2^n equivalent corner cells each, until the number of remaining cells is congruent to k (mod 2*n). The resulting polyhypercube is still connected. Then add sets of 2*n spike cells until the total number of cells is equal to k. This proves the second part. The bound k >= (4*n-1)^n resulting from this construction is far from optimal.

Row sums are the powers of 3 (A000244).

T(n,k) equals the number of n-length words on {0,1,...,13} having n-k zeros. - Milan Janjic, Jul 24 2015

Row sums are:

{0, 3, 9, 99, 1161, 4617, 55323, 663633, 2654289, 31850739, 382206681...}

These are a sequence of Infinite sums that give A142458.

Even terms are factored by (1+2*n) which is the MacMahon (1+2*n)^k,but the polynomials seem fundamental

other than that.

A060187 MacMahon gives A013609 Triangle of coefficients in expansion of (1 + 2x)^n.

I looked for a simple infinite sum for the {1,8,1} and failed.

This reasoning comes from finding that the general z Transform polynomials are

related to the Eulerian: in fact this type of Eulerian polynomials A008292 gives (1+n)^k binomial.

The polynomials given here form a set of infinite sum sequences.

Row sums are 3^n - 1 + 0^n.

Triangle of coefficients in expansion of (1+2*x)^n - 1 + 0^n .

A117317 (A):

1

2 1

4 5 1

8 16 9 1

16 44 41 14 1

32 112 146 85 20 1

64 272 456 377 155 27 1

have for their columns successive signatures

(2) (4,-4) (6,-12,8) (8,-24, 32, -16) (10,-40,80,-80,32) i.e. a(n).

Take based on abs(A133156) (B):

1

2 0

4 1 0

8 4 0 0

16 12 1 0 0

32 32 6 0 0 0

64 80 24 1 0 0 0.

The recurrences of successive columns are also a(n). a(n) columns: A005843(n+1), A046092(n+1), A130809, A130810, A130811, A130812, A130813.

A053220 + A001787 = A014480.

Let n be the dimension of the cubic array.

Let m be the "placement depth" of the cell within the array. m = (number of horizontal or vertical neighbors)-n. 0 <= m <= n.

Let T(n,m) represent the number of neighbors (horizontally, vertically, or diagonally) a cell has in an n-dimensional cube that has at least 3^n cells.

The sequence forms a triangle structure similar to Pascal’s triangle: T(0,0) in row one, T(1,0), T(1,1) in row two, etc.

The triangle in A094615 is a subtriangle. - Philippe Deléham, Oct 31 2013

In a finite n-dimensional hypercube lattice, the sequence gives the number of nodes situated at a Chebyshev distance of 1 for a node, situated on an m-cube bound, which is not on an (m-1)-cube bound. The number of m-cube bounds for n-cube is given by A013609. In cellular automata theory, the cell surrounding with Chebyshev distance 1 is called Moore's neighborhood. For von Neumann neighborhood (with Manhattan distance 1), an analogous sequence is represented by A051162. - Dmitry Zaitsev, Oct 22 2015

Number of ternary strings of length n that have the same number or more 0's than the combined number of 1's and 2's. For example, a(4) = 33 since the strings are (number of permutations in parentheses): 0000 (1), 0001 (4), 0002 (4), 0011 (6), 0022 (6), 0012 (12). - Enrique Navarrete, Aug 14 2025

The numbers in rows of the triangle are along a "third layer" skew diagonals pointing top-right in center-justified triangle given in A013609 ((1+2*x)^n) and along a "third layer" skew diagonals pointing top-left in center-justified triangle given in A038207 ((2+x)^n), see links. (Note: First layer skew diagonals in center-justified triangles of coefficients in expansions of (1+2*x)^n and (2+x)^n are given in A128099 and A207538 respectively.)

The coefficients in the expansion of 1/(1-x-2*x^4) are given by the sequence generated by the row sums.

The row sums give A052942.

If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 1.543689012692076... (A256099: Decimal expansion of the real root of a cubic used by Omar Khayyám in a geometrical problem), when n approaches infinity.