A268759 - OEIS

A268759

Triangle T(n,k) read by rows: T(n,k) = (1/4)*(1 + k)*(2 + k)*(k - n)*(1 + k - n).

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

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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

Table of n, a(n) for n=0..70.

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;

1, 0, 0;

3, 3, 0, 0;