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 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) = = = - = - . In the Dirac notation, we write elements m_{ij} of matrix M as =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 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, 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 Cf. A114327, A094053, A141387. Sequence in context: A298629 A298461 A299334 * A298841 A299602 A299554 Adjacent sequences: A268756 A268757 A268758 * A268760 A268761 A268762 KEYWORD nonn,tabl AUTHOR Bradley Klee, Feb 20 2016 STATUS approved

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