A119937 Triangle of numbers related to the spectrum of the hydrogen (H) atom.

3, 32, 5, 135, 27, 7, 3456, 756, 256, 81, 3500, 800, 300, 125, 44, 172800, 40500, 16000, 7425, 3456, 1300, 694575, 165375, 67375, 33075, 17199, 8575, 3375, 6272000, 1509200, 627200, 318500, 175616, 98000, 51200

Offset: 2

Author

Wolfdieter Lang, Jul 20 2006

Keywords

• nonn
• easy
• tabl

Comments

The rational number triangle r(m,n):=A120072(m,n)/A120073(m,n), used to compute the spectral series of the hydrogen atom, is mapped to this nonnegative number triangle by multiplying the least common multiples (LCM) for each row m.

Examples

			[3]; [32,5]; [135,27,7]; [3456,756,256,81]; [3500,800,300,125,44]; ...
		

Links

• W. Lang: First ten rows.

Crossrefs

The LCM sequence which has been used here is [4, 36, 144, 3600, 3600, 176400, 705600, 6350400, 6350400, 768398400, ...] = A051418(m) = (A003418(m))^2 = (2*A025555(m-1))^2, m >= 2.

The row sums give A119938.

Formula

a(m,n) = r(m,n)*lcm_{k=1..m-1} seq(r(m,k)) with r(m,n) = 1/n^2 - 1/m^2 = A120072(m,n)/A120073(m,n), m >= 2, n = 1..m-1.