# 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: A_{0}=0, A_{1}=0, A_{2}=0, A_{3}=0, A_{4}=1,
A_{5}=10, A_{6}=50, A_{7}=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, A_{5} =
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 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 51From 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.

A_{1} = A_{2} = A_{3} = 0

A_{4} = 1

A_{5} = 10 = 2 5

A_{6} = 50 = 2 5^{2}

A_{7} = 175 = 5^{2} 7

A_{8} = 490 = 2 5 7^{2}

A_{9} = 1176 = 2^{3} 3 7^{2}

A_{10} = 2520 = 2^{3} 3^{2} 5 7

A_{11} = 4950 = 2 3^{2} 5^{2} 11

A_{12} = 9075 = 3 5^{2} 11^{2}

A_{13} = 15730 = 2 5 11^{2} 13

A_{14} = 26026 = 2 7 11 13^{2}

A_{15} = 41405 = 5 7^{2} 13^{2}

A_{16} = 63700 = 2^{2} 5^{2} 7^{2} 13

A_{17} = 95200 = 2^{5} 5^{2} 7 17

A_{18} = 138720 = 2^{5} 3 5 17^{2}

A_{19} = 197676 = 2^{2} 3^{2} 17^{2} 19

A_{20} = 276165 = 3^{2} 5 17 19^{2}

A_{21} = 379050 = 2 3 5^{2} 7 19^{2}

A_{22} = 512050 = 2 5^{2} 7^{2} 11 19

A_{23} = 681835 = 5 7^{2} 11^{2} 23

A_{24} = 896126 = 2 7 11^{2} 23^{2}

A_{25} = 1163800 = 2^{3} 5^{2} 11 23^{2}

A_{26} = 1495000 = 2^{3} 5^{4} 13 23

A_{27} = 1901250 = 2 3^{2} 5^{4} 13^{2}

Some other sequences are discussed here.

This page was written in the "embarrassingly readable" markup language RHTF, and was last updated on 2022 Mar 28. s.30