A362534 Numerators of the ratio of the symmetry-constrained bound to the adiabatic bound on polarization transfer in AXn spin-1/2 systems.

1, 1, 6, 6, 15, 15, 140, 140, 315, 315, 1386, 1386, 3003, 3003, 51480, 51480, 109395, 109395, 92378, 92378, 969969, 969969, 2704156, 2704156, 16900975, 16900975, 70204050, 70204050, 145422675, 145422675, 4808643120, 4808643120, 9917826435, 9917826435, 40838108850, 40838108850

Offset: 1

Author

Mohamed Sabba, Apr 24 2023

Keywords

• nonn
• frac

Comments

In spin physics and NMR, these numbers appear as the numerators of the ratio of different classes of upper bounds on the transfer of z-magnetization in AXn spin systems from a group of spin-1/2 nuclei Xn to a single spin-1/2 nucleus A.

The symmetry-constrained upper bounds are given by the function f(n):

(1) for even n, f(n) = (2^(1-n))*n*binomial(n-1, n/2)

(2) for odd n, f(n) = (2^(1-n))*n*binomial(n-1, (n-1)/2)

The adiabatic bounds are given by the function g(n):

(3) for even n, g(n) = 2*(1-(2^(-n))*binomial(n, n/2))

(4) for odd n, g(n) = 2*(1-(2^(-n))*binomial(n, (n-1)/2))

Where we have the relation:

(5) g(n) = 2*(1 - f(n+1)/(n+1))

The sequence a(n) is defined as the numerator of f(n)/g(n):

(6) a(n) = numerator(f(n)/g(n))

(7) for even n, f(n)/g(n) = (n/2)/(2^(n)*binomial(n, n/2)^(-1) - 1)

(8) for odd n, f(n)/g(n) = ((n+1)/2)/(2^(n)*binomial(n, (n+1)/2)^(-1) - 1)

The first few values of the upper symmetry-constrained bounds f(n) are {1, 1, 3/2, 3/2, 15/8, 15/8, 35/16, 35/16, 315/128, 315/128, ...} which appears to be related to A086116 and A001803.

The first few values of the upper adiabatic bounds g(n) are {1, 1, 5/4, 5/4, 11/8, 11/8, 93/64, 93/64, 193/128, 193/128, ...} which appears to be related to A141244 and A120778.

The first few values of f(n)/g(n) are {1, 1, 6/5, 6/5, 15/11, 15/11, 140/93, 140/93, 315/193, 315/193, ...}

Conjecture: the numerator of g(n) is the denominator of f(n)/g(n).

Links

• G. C. Chingas, A. N. Garroway, W. B. Moniz, and R. D. Bertrand, Adiabatic J cross-polarization in liquids for signal enhancement in NMR, Journal of Chemical Physics, 102:8 (1980), 2526-2528 (page 1, equation 2 gives an expression for the adiabatic bounds).
• Malcolm H. Levitt, Symmetry constraints on spin dynamics: Application to hyperpolarized NMR, Journal of Chemical Physics, 102:8 (2016).
• Ole W. Sørensen, A universal bound on spin dynamics, Journal of Magnetic Resonance, 262 (1990) (appendix gives proof of the expression for symmetry constrained bounds).

Crossrefs

Cf. A001803, A086116, A120778, A141244 (denominators but shifted).

Programs

Mathematica

Table[Numerator[Ceiling[n/2]  (2^n Binomial[n, Ceiling[n/2]]^-1 - 1 )^-1], {n, 1, 20}]

PARI

a(n) = numerator(ceil(n/2)/(2^(n)*binomial(n,ceil(n/2))^(-1) - 1)); \\ Michel Marcus, Apr 25 2023

Formula

a(n) = numerator(ceiling(n/2)/(2^(n)*binomial(n,ceiling(n/2))^(-1) - 1)).