Sequence A006542, C(n,3)C(n-1,3)/4  

This sequence, Sloane's A006542, is a 6-dimensional figurate sequence and the 4th 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 AN=C(N,3)×C(N-1,3)/4, where C(N,M) are the binomial coefficents found in Pascal's triangle: the Mth number in row N. So, for example, A5 = C(5,3)×C(4,3)/4 = 10×4/4 = 10.

This sequence appears as the 4th 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 centered 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 center, 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 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.



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