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

1,3

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

Table of n, a(n) for n=1..36.

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).

Formula

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

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A168414 A266223 A256675 * A290931 A257372 A058563

Adjacent sequences: A362531 A362532 A362533 * A362535 A362536 A362537

Keyword

nonn,frac

Author

Mohamed Sabba, Apr 24 2023

Status

approved