Sequence A006542, C(n,3)C(n-1,3)/4

This sequence, Sloane's A006542, is a 6-dimensional figurate sequence and the 4^th diagonal of the Narayana triangle.

The sequence begins: A₀=0, A₁=0, A₂=0, A₃=0, A₄=1, A₅=10, A₆=50, A₇=175; and continues: 490, 1176, 2520, 4950, 9075, 15730, 26026, 41405, 63700, 95200, 138720, 197676, 276165, 379050, 512050, 681835, 896126, 1163800, 1495000, 1901250, ...

The simplest direct formula for a term in the sequence is A_N=C(N,3)×C(N-1,3)/4, where C(N,M) are the binomial coefficents found in Pascal's triangle: the M^th number in row N. So, for example, A₅ = C(5,3)×C(4,3)/4 = 10×4/4 = 10.

This sequence appears as the 4^th diagonal of the Catalan triangle (Narayana numbers) shown here and also discussed below.

Geometric Interpretation

To visualise this sequence as a geometric figure, we start with the centreed pentagonal numbers (Sloane's A005891). This sequence starts: 1, 6, 16, 31, 51, 76, 106, 141, 181, 226, 276, ... and can be visualised as a set of concentric rings of pentagonal shape, with one in the very centre, 5 in the first ring, 10 in the next ring, and so on:

o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o 1 6 16 31 51

From this sequence, we construct a sequence of 5-sided pyramids. The pyramid has a pentagonal base, with objects arranged like one of the figures above, and a number of layers on top of it, each with a progressively smaller pentagonal arrangement. It's hard to do with round objects like apples or oranges, because with each layer you have to balance each item perfectly on top of a single item on the layer below, but it can be done with cylinders (cans) or with blocks. The sequence we get this way is: 1, 1+6, 1+6+16, 1+6+16+31, ... which produces 1, 7, 23, 54, 105, 181, 287, 428, 609, 835, 1111, 1442, ... (Sloane's A004068).

Next, imagine constructing a 4-dimensional stack of objects in which each layer is a 5-sided pyramid. In 4 dimensions, such a figure would have 7 "faces" (which in 4-dimensional geometry are usually called cells): a "base" cell which is a 5-sided pyramid, and 6 more "side" cells corresponding to the 6 faces of the 5-sided pyramid, i.e. a 5-sided-pyramidal cell and five tetrahedral cells. Counting the individual blocks that are needed to create such an arrangement, we get: 1, 1+7, 1+7+23, 1+7+23+54, ... which produces 1, 8, 31, 85, 190, 371, 658, 1086, 1695, 2530, 3641, ... (Sloane's A006322).

Now we perform the same process again, stacking 4-dimensional layers to create 5-dimensional figures. Here it is pretty difficult to describe what it looks like geometrically, except that it is bounded by three 4-dimensional solids resembling the figures from the previous sequence, and five simplexes (hyper-tetrahedra). The sequence of numbers we get is: 1, 1+8, 1+8+31, 1+8+31+85, ... which produces 1, 9, 40, 125, 315, 686, 1344, 2430, 4125, 6655, ... (Sloane's A006414).

We're almost done. Perform the same stacking process one more time, and we produce a sequence of 6-dimensional figures. The sequence is: 1, 1+9, 1+9+40, 1+9+40+125, ... which produces 1, 10, 50, 175, 490, 1176, 2520, 4950, ... which is Sloane's A006542 and the subject of this page.

Factorisation

Because the numbers come from a product of binomial coefficients, their factors are all relatively small, and tend to be fairly well distributed.

A₁ = A₂ = A₃ = 0
A₄ = 1
A₅ = 10 = 2 5
A₆ = 50 = 2 5²
A₇ = 175 = 5² 7
A₈ = 490 = 2 5 7²
A₉ = 1176 = 2³ 3 7²
A₁₀ = 2520 = 2³ 3² 5 7
A₁₁ = 4950 = 2 3² 5² 11
A₁₂ = 9075 = 3 5² 11²
A₁₃ = 15730 = 2 5 11² 13
A₁₄ = 26026 = 2 7 11 13²
A₁₅ = 41405 = 5 7² 13²
A₁₆ = 63700 = 2² 5² 7² 13
A₁₇ = 95200 = 2⁵ 5² 7 17
A₁₈ = 138720 = 2⁵ 3 5 17²
A₁₉ = 197676 = 2² 3² 17² 19
A₂₀ = 276165 = 3² 5 17 19²
A₂₁ = 379050 = 2 3 5² 7 19²
A₂₂ = 512050 = 2 5² 7² 11 19
A₂₃ = 681835 = 5 7² 11² 23
A₂₄ = 896126 = 2 7 11² 23²
A₂₅ = 1163800 = 2³ 5² 11 23²
A₂₆ = 1495000 = 2³ 5⁴ 13 23
A₂₇ = 1901250 = 2 3² 5⁴ 13²

Some other sequences are discussed here.

This page was written in the "embarrassingly readable" markup language RHTF, and was last updated on 2024 Oct 02.

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