login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A268759 Triangle T(n,k) read by rows: T(n,k) = (1/4)*(1 + k)*(2 + k)*(k - n)*(1 + k - n). 3
0, 0, 0, 1, 0, 0, 3, 3, 0, 0, 6, 9, 6, 0, 0, 10, 18, 18, 10, 0, 0, 15, 30, 36, 30, 15, 0, 0, 21, 45, 60, 60, 45, 21, 0, 0, 28, 63, 90, 100, 90, 63, 28, 0, 0, 36, 84, 126, 150, 150, 126, 84, 36, 0, 0, 45, 108, 168, 210, 225, 210, 168, 108, 45, 0, 0, 55, 135, 216, 280, 315 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,7
COMMENTS
Off-diagonal elements of angular momentum matrices J_1^2 and J_2^2.
Construct the infinite-dimensional matrix representation of angular momentum operators (J_1,J_2,J_3) in the block-diagonal, Jordan-Schwinger form (cf. Harter, Klee, Schwinger). The triangle terms T(n,k) satisfy:(1/2)T(n,k)^(1/2) = <n(n+1)/2+k+1|J_1^2|n(n+1)/2+k+3> = <n(n+1)/2+k+3|J_1^2|n(n+1)/2+k+1> = - <n(n+1)/2+k+1|J_2^2|n(n+1)/2+k+3> = - <n(n+1)/2+k+3|J_2^2|n(n+1)/2+k+1>. In the Dirac notation, we write elements m_{ij} of matrix M as <i|M|j>=m_{ij}. Matrices for J_1^2 and J_2^2 are sparse. These equalities and the central-diagonal equalities of A141387 determine the only nonzero entries.
Notice that a(n) = T(n,k) is always a multiple of the triangular numbers, up to an offset. Conjecture: the triangle tabulating matrix elements <n(n+1)/2+k+1|J_1^p|n(n+1)/2+k+p+1> is determined entirely by the coefficients: binomial(n,p) (cf. A094053). Various sequences along the diagonals of matrix J_1^p lead to other numbers with geometric interpretations (Cf. A000567, A100165).
LINKS
W. Harter, Principles of Symmetry, Dynamics, Spectroscopy, Wiley, 1993, Ch. 5, page 345-346.
B. Klee, Quantum Angular Momentum Matrices, Wolfram Demonstrations Project, 2016.
J. Schwinger, On Angular Momentum , Cambridge: Harvard University, Nuclear Development Associates, Inc., 1952.
FORMULA
T(n,k) = (1/4)*(1 + k)*(2 + k)*(k - n)*(1 + k - n).
G.f.: x^2/((1-x)^3(1-x*y)^3)
EXAMPLE
0;
0, 0;
1, 0, 0;
3, 3, 0, 0;
6, 9, 6, 0, 0;
10, 18, 18, 10, 0, 0;
15, 30, 36, 30, 15, 0, 0;
...
MATHEMATICA
Flatten[Table[(1/4) (1 + k) (2 + k) (k - n) (1 + k - n), {n, 0, 10, 1}, {k, 0, n, 1}]]
CROSSREFS
Sequence in context: A298629 A298461 A299334 * A298841 A299602 A299554
KEYWORD
nonn,tabl
AUTHOR
Bradley Klee, Feb 20 2016
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified April 20 12:36 EDT 2024. Contains 371844 sequences. (Running on oeis4.)