A098737 Triangle read by rows: number of triangles formed by lines from two vertices of a triangle to the opposite side that segment the opposite sides into m and n segments. Since f(m,n) = f(n,m), it suffices to give the results in a triangular table.

1, 3, 8, 6, 15, 27, 10, 24, 42, 64, 15, 35, 60, 90, 125, 21, 48, 81, 120, 165, 216, 28, 63, 105, 154, 210, 273, 343, 36, 80, 132, 192, 260, 336, 420, 512, 45, 99, 162, 234, 315, 405, 504, 612, 729, 55, 120, 195, 280, 375, 480, 595, 720, 855, 1000, 66, 143, 231, 330

Offset: 1

Author

Eugene McDonnell (eemcd(AT)mac.com), Oct 29 2004

Keywords

• easy
• nonn
• tabl

Comments

Frank Buss gave this as a puzzle; K. L. Metlov solved it, submitting his result in the J language created by Kenneth Iverson. The program given below is only five tokens long. J defines a series of three functions to be a "fork" defined by x (f g h ) y = (x f y) g (f h y) - a generalization of the usual mathematical practice of writing (f + g) y to mean (f y) + (g y). J also has a primitive "half" and has a dummy function "cap" whose purpose is to permit more forks to be written. 3 (* * +) 5 is thus (3 * 5) * (3 + 5) or 120. cap half 3 (* * +) 5 is thus 60.

This sequence is the dimensions of the various irreducible representations of SU(3). In the language of physics, the integers m and n are one more than the numbers of quarks or antiquarks, respectively, that label the representation. - Alex Meiburg, Dec 13 2020 =

Comment on the previous one: D(n, m) = f(m+1, n+1) = (n+1)*(m+1)*(n+m+2), for 0 <= m <= n, (given as array D(n,m) as example in A212331) is the dimension of the irreducible SU(3) multiplet (n, m), denoted also by D(n, m). The multiplet (m, n) is denoted also by a bar over D(n, m). The irreducuble tensor t(n, m) is symmetric in n upper indices from {1,2,3}, symmetric in m lower indices, and traceless in every pair of an upper and a lower index. See the Coleman reference for a derivation. - Wolfdieter Lang, Dec 18 2020

Examples

			f(3, 5) is 60, from 1/2 * (3 * 5) * (3 + 5) or 1/2 * 15 * 8.
The triangle f(m, n) starts:
m\n     1   2   3   4   5   6   7   8   9   10   11 ...
1:      1
2:      3   8
3:      6  15  27
4:     10  24  42  64
5:     15  35  60  90 125
6:     21  48  81 120 165 216
7:     28  63 105 154 210 273 343
8:     36  80 132 192 260 336 420 512
9:     45  99 162 234 315 405 504 612 729
10:    55 120 195 280 375 480 595 720 855 1000
11:    66 143 231 330 440 561 693 836 990 1155 1331
... reformatted and extended by _Wolfdieter Lang_, Dec 18 2020
		

References

• Sidney Coleman, Quantum Field Theory, Eds. Bryan Gin-ge Chen et al., World Scientific, 2019, eq. (37.8), p. 799.

Links

• Wikipedia, Clebsch-Gordan coefficients for SU(3)

Crossrefs

Cf. A000217, A005563, A140091, A067728, A212331, A140681 (columns), A000578, A059270, A331433 (diagonals).

(diagonal).

See also A107985, A212331 (array as example).

Programs

J

cap half * * +

Mathematica

t[m_, n_] := (m*n)(m + n)/2; Flatten[ Table[ t[m, n], {m, 10}, {n, m}]] (* Robert G. Wilson v, Nov 04 2004 *)

Formula

f(m, n) = 1/2 * (m * n) * (m + n).

G.f.: x*y*(1 + 4*x*y + x^2*(y - 9)*y - 3*x^3*(y - 1)*y + 3*x^4*y^2)/((1 - x)^3*(1 - x*y)^4). - Stefano Spezia, Oct 01 2023

Extensions

More terms from Robert G. Wilson v, Nov 04 2004