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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).

%I A268759 #36 Nov 08 2016 21:02:47
%S A268759 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,
%T A268759 45,60,60,45,21,0,0,28,63,90,100,90,63,28,0,0,36,84,126,150,150,126,
%U A268759 84,36,0,0,45,108,168,210,225,210,168,108,45,0,0,55,135,216,280,315
%N A268759 Triangle T(n,k) read by rows: T(n,k) = (1/4)*(1 + k)*(2 + k)*(k - n)*(1 + k - n).
%C A268759 Off-diagonal elements of angular momentum matrices J_1^2 and J_2^2.
%C A268759 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.
%C A268759 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).
%H A268759 W. Harter, <a href="http://www.uark.edu/ua/modphys/markup/PSDS_Info.html/">Principles of Symmetry, Dynamics, Spectroscopy</a>, Wiley, 1993, Ch. 5, page 345-346.
%H A268759 B. Klee, <a href="http://demonstrations.wolfram.com/QuantumAngularMomentumMatrices/">Quantum Angular Momentum Matrices</a>, Wolfram Demonstrations Project, 2016.
%H A268759 J. Schwinger, <a href="http://www.ifi.unicamp.br/~cabrera/teaching/paper_schwinger.pdf"> On Angular Momentum </a>, Cambridge: Harvard University, Nuclear Development Associates, Inc., 1952.
%F A268759 T(n,k) = (1/4)*(1 + k)*(2 + k)*(k - n)*(1 + k - n).
%F A268759 G.f.: x^2/((1-x)^3(1-x*y)^3)
%e A268759 0;
%e A268759 0,  0;
%e A268759 1,  0,  0;
%e A268759 3,  3,  0,  0;
%e A268759 6,  9,  6,  0,  0;
%e A268759 10, 18, 18, 10, 0,  0;
%e A268759 15, 30, 36, 30, 15, 0, 0;
%e A268759 ...
%t A268759 Flatten[Table[(1/4) (1 + k) (2 + k) (k - n) (1 + k - n), {n, 0, 10, 1}, {k, 0, n, 1}]]
%Y A268759 Cf. A114327, A094053, A141387.
%K A268759 nonn,tabl
%O A268759 0,7
%A A268759 _Bradley Klee_, Feb 20 2016