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 A268533 Pascal's difference pyramid read first by blocks and then by rows: T(n,k,m) = 1/(m!) (d/dx)^m((1 - x)^k*(1 + x)^(n - k))|_{x=0}. 2
 1, 1, 1, 1, -1, 1, 2, 1, 1, 0, -1, 1, -2, 1, 1, 3, 3, 1, 1, 1, -1, -1, 1, -1, -1, 1, 1, -3, 3, -1, 1, 4, 6, 4, 1, 1, 2, 0, -2, -1, 1, 0, -2, 0, 1, 1, -2, 0, 2, -1, 1, -4, 6, -4, 1, 1, 5, 10, 10, 5, 1, 1, 3, 2, -2, -3, -1, 1, 1, -2, -2, 1, 1, 1, -1, -2, 2, 1, -1, 1, -3, 2, 2, -3, 1, 1, -5, 10, -10, 5, -1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,7 COMMENTS T(n,k,m) is a pyramidal stack of (n+1)-by-(n+1) dimensional matrices, or an infinite-dimensional matrix in block-diagonal form ( see examples ). Define triangular slices T_x(i,j) = T(2x+i,x,x+j) with i in {0,1,...} and j in {0,1,... i}. T_0 is Pascal's triangle, and it appears that T_{x} is a triangle of first differences T_{x}(i,j) = T_{x-1}(i+1,j+1)-T_{x-1}(i+1,j) (Cf. A007318, A214292). The so-called "quantum Pascal's pyramid", denoted QT(n,k,m), is obtained from Pascal's pyramid by a complexification of matrix elements: QT(n,k,m) = (-1)^(3m/2) T(n,k,m). QT(n,k,m) affects a Hermite-Cartesian ( Cf. A066325 ) to Laguerre-polar change of coordinates ( see examples ). Row reversal is complex conjugation: QT(n,n-k,m)=QT(n,k,m)*. To construct the "normalized quantum Pascal's pyramid", NQT(n,k,m), we need normalization numerators, NumT(n,k,m) as in A269301,  and denominators, DenT(n,k,m) as in A269302; then, NQT(n,k,m)= sqrt[ NumT(n,k,m) / DenT(n,k,m) ] QT(n,k,m). In the context of physics NQT(n,k,m) acting as matrix conjugation affects a cyclic permutation of the infinite-dimensional generators of rotation, so NQT(n,k,m) is essentially equivalent to an infinite-dimensional rotation with (z,y,z) Euler angles (0,Pi/2,Pi/2) (Harter, Klee, see examples). Normalization or no, Pascal's pyramid also arises in laser optics (Allen et al.) as the paraxial wave equation often admits a useful analogy to the SchrÃ¶dinger equation for the two-dimensional isotropic quantum harmonic oscillator. REFERENCES L. Allen, S.M. Barnett, and M.J. Padgett, Optical angular momentum, Institute of Physics Publishing, Bristol, 2003. LINKS L. Allen et al., Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes, Physical Review A, 45 (1992), 8185-8190. W. Harter, Principles of Symmetry, Dynamics, Spectroscopy, Wiley, 1993, Ch. 5, page 345-348. B. Klee, Quantum Angular Momentum Matrices, Wolfram Demonstrations Project, 2016. FORMULA T(n,k,m) = 1/(m!) (d/dx)^m((1 - x)^k*(1 + x)^(n - k))|_{x=0}. EXAMPLE First few blocks: 1 .  1,  1 .  1, -1 . . . . .  1,  2,  1 . . . . .  1,  0, -1 . . . . .  1, -2,  1 . . . . . . . . . . .  1,  3,  3,  1 Second triangle . . .  1,  1, -1, -1 slice, T_1: . . . . .  1, -1, -1,  1 0 . . . . . . . . . .  1, -3,  3, -1 1  -1 . . . . . . . . . . . . . . . .  1,  4,  6,  4,  1 2   0  -2 . . . . . . . . . . . . . .  1,  2,  0, -2, -1 3,  2, -2, -3 . . . . . . . . . . . .  1,  0, -2,  0,  1 4,  5,  0, -5, -4 . . . . . . . . . .  1, -2,  0,  2, -1 5,  9,  5, -5, -9, -5 . . . . . . . .  1, -4,  6, -4,  1 n=2 Cartesian/Polar coordinate change using quantum Pascal's pyramid: | 1  -2 i  -1 |   | y^2 - 1 |    | - (r exp[ I \phi])^2 | | 1   0     1 | * |   x*y   | =  |      r^2  -  2       | | 1   2 i  -1 |   | x^2 - 1 |    | - (r exp[-I \phi])^2 | When: x = r cos[\phi], y= r sin[\phi]. Permutation of Pauli Matrices, \sigma_i, using normalized quantum Pascal's pyramid:                   | 1  -i | R = (1/sqrt[2]) * | 1   i | Then, R * \sigma_j * R^{\dagger} = \sigma_{pi(j)}, where pi(j) is a cyclic permutation: { 1 -> 2, 2 -> 3, 3 -> 1 }. MATHEMATICA PascalsPyramid[Block_] := Outer[Simplify[Function[{n, k, m}, 1/(m!)(D[(1 - x)^k*(1 + x)^(n - k), {x, m}] /. x -> 0)][Block, #1, #2]] &, Range[0, Block], Range[0, Block]]; PascalsPyramid /@ Range[0, 10] CROSSREFS Cf. A007318, A214292, A269301, A269302. Sequence in context: A173432 A101675 A051764 * A275849 A025906 A213369 Adjacent sequences:  A268530 A268531 A268532 * A268534 A268535 A268536 KEYWORD sign AUTHOR Bradley Klee, Feb 22 2016 STATUS approved

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Last modified January 27 02:46 EST 2020. Contains 331291 sequences. (Running on oeis4.)