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A268759
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Triangle T(n,k) read by rows: T(n,k) = (1/4)*(1 + k)*(2 + k)*(k - n)*(1 + k - n).
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3
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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
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0,7
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COMMENTS
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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).
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LINKS
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J. Schwinger, On Angular Momentum , Cambridge: Harvard University, Nuclear Development Associates, Inc., 1952.
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FORMULA
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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)
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EXAMPLE
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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;
...
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MATHEMATICA
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Flatten[Table[(1/4) (1 + k) (2 + k) (k - n) (1 + k - n), {n, 0, 10, 1}, {k, 0, n, 1}]]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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