Pentatope number

A pentatope number is a number in the fifth cell of any row of Pascal's triangle starting with the 5-term row 1 4 6 4 1 either from left to right or from right to left.

The first few numbers of this kind are :

1, 5, 15, 35, 70, 126, 210, 330, 495, 715, 1001, 1365 (sequence A000332 in the OEIS)

Pentatope numbers belong in the class of figurate numbers, which can be represented as regular, discrete geometric patterns.^[1] The formula for the nth pentatopic number is:

${n + 3 \choose 4} = \frac{n(n+1)(n+2)(n+3)}{24} = {n^{\overline 4} \over 4!}.$

Two of every three pentatope numbers are also pentagonal numbers. To be precise, the (3k − 2)th pentatope number is always the ((3k² − k)/2)th pentagonal number and the (3k − 1)th pentatope number is always the ((3k² + k)/2)th pentagonal number. The 3kth pentatope number is the generalized pentagonal number obtained by taking the negative index −(3k² + k)/2 in the formula for pentagonal numbers. (These expressions always give integers).^[2]

The infinite sum of the reciprocals of all pentatopal numbers is $4 \over 3$.^[3] This can be derived using telescoping series.

$\sum_{n=1}^\infty {4! \over {n(n+1)(n+2)(n+3)}} = {4 \over 3}$

Pentatopal numbers can also be represented as the sum of the first n tetrahedral numbers.^[2]

Test for pentatope numbers

$\frac{\sqrt{24n+1}+1}{2}$ is triangular number.

then : ${\displaystyle 8\times {\frac {{\sqrt {24n+1}}+1}{2}}+1}$ is perfect square.

References

1. Deza, Elena; Deza, M. (2012), "3.1 Pentatope numbers and their multidimensional analogues", Figurate Numbers, World Scientific, p. 162, ISBN 9789814355483
2. "Sloane's A000332". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
3. Rockett, Andrew M. (1981), "Sums of the inverses of binomial coefficients" (PDF), Fibonacci Quarterly, 19 (5): 433–437. Theorem 2, p. 435.