A269301 -id:A269301 - cp's OEIS frontend

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}.

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

Offset: 0

Bradley Klee, Feb 22 2016

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T(n,k,m) is a pyramidal stack of (n+1) X (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) effects 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 effects 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 Schrödinger equation for the two-dimensional isotropic quantum harmonic oscillator.

			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 }.

L. Allen, S. M. Barnett, and M. J. Padgett, Optical angular momentum, Institute of Physics Publishing, Bristol, 2003.

L. Allen et al., Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes, Physical Review A, 45 (1992), 8185-8190.
William G. Harter, Principles of Symmetry, Dynamics, Spectroscopy, Wiley, 1993, Ch. 5, page 345-348.
Brad Klee, Quantum Angular Momentum Matrices, Wolfram Demonstrations Project, 2016.
Mohamed Sabba, A quantum Pascal pyramid and an extended de Moivre-Laplace theorem, arXiv:2404.03560 [quant-ph], 2024. See pp. 1-2.

Cf. A007318, A214292, A269301, A269302.

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]

T(n,k,m) = (1/(m!)) * (d/dx)^m((1-x)^k*(1+x)^(n-k))|_{x=0}.

A269302 Normalization coefficients for quantum Pascal's pyramid, denominators of: T(n,k,m) = ((n - m)! m!)/(2^n (n - k)! k!).

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 8, 4, 2, 4, 2, 4, 8, 4, 8, 24, 24, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 24, 24, 8, 16, 64, 96, 64, 16, 4, 16, 24, 16, 4, 8, 32, 16, 32, 8, 4, 16, 24, 16, 4, 16, 64, 96, 64, 16, 32, 160, 320, 320, 160, 32, 32, 32, 64, 64, 32, 32, 16, 16, 32, 32, 16, 16, 16, 16, 32, 32, 16, 16, 32, 32, 64, 64, 32, 32, 32, 160, 320, 320, 160, 32

Offset: 0

Bradley Klee, Feb 22 2016

Read by block by row, i.e., a( x(n,k,m) ) have x(n,k,m) = ( sum_{i=0}^n i^2 ) + k ( n + 1 ) + m and (n,k,m) >= 0. See comments in A268533 for relevance.

			First few blocks:
1
. . 2, 2
. . 2, 2
. . . . .  4, 8, 4
. . . . .  2, 4, 2
. . . . .  4, 8, 4

Numerators: A269301. Cf. A268533.

NormFrac[Block_] :=
Outer[Function[{n, k, m}, ((n - m)! m!)/(2^n (n - k)! k!)][
    Block, #1, #2] &, Range[0, Block], Range[0, Block], 1]; Flatten[
Denominator[NormFrac[#]] & /@ Range[0, 5]]

T(n,k,m) = Denominator[((n - m)! m!)/(2^n (n - k)! k!)]

cp's OEIS Frontend

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}.

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