cp's OEIS Frontend

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

%I A362534 #24 Jun 11 2023 12:13:51
%S A362534 1,1,6,6,15,15,140,140,315,315,1386,1386,3003,3003,51480,51480,109395,
%T A362534 109395,92378,92378,969969,969969,2704156,2704156,16900975,16900975,
%U A362534 70204050,70204050,145422675,145422675,4808643120,4808643120,9917826435,9917826435,40838108850,40838108850
%N A362534 Numerators of the ratio of the symmetry-constrained bound to the adiabatic bound on polarization transfer in AXn spin-1/2 systems.
%C A362534 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.
%C A362534 The symmetry-constrained upper bounds are given by the function f(n):
%C A362534 (1) for even n, f(n) = (2^(1-n))*n*binomial(n-1, n/2)
%C A362534 (2) for odd n, f(n) = (2^(1-n))*n*binomial(n-1, (n-1)/2)
%C A362534 The adiabatic bounds are given by the function g(n):
%C A362534 (3) for even n, g(n) = 2*(1-(2^(-n))*binomial(n, n/2))
%C A362534 (4) for odd n, g(n) = 2*(1-(2^(-n))*binomial(n, (n-1)/2))
%C A362534 Where we have the relation:
%C A362534 (5) g(n) = 2*(1 - f(n+1)/(n+1))
%C A362534 The sequence a(n) is defined as the numerator of f(n)/g(n):
%C A362534 (6) a(n) = numerator(f(n)/g(n))
%C A362534 (7) for even n, f(n)/g(n) = (n/2)/(2^(n)*binomial(n, n/2)^(-1) - 1)
%C A362534 (8) for odd n, f(n)/g(n) = ((n+1)/2)/(2^(n)*binomial(n, (n+1)/2)^(-1) - 1)
%C A362534 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.
%C A362534 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.
%C A362534 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, ...}
%C A362534 Conjecture: the numerator of g(n) is the denominator of f(n)/g(n).
%H A362534 G. C. Chingas, A. N. Garroway, W. B. Moniz, and R. D. Bertrand, <a href="https://doi.org/10.1021/ja00528a002">Adiabatic J cross-polarization in liquids for signal enhancement in NMR</a>, Journal of Chemical Physics, 102:8 (1980), 2526-2528 (page 1, equation 2 gives an expression for the adiabatic bounds).
%H A362534 Malcolm H. Levitt, <a href="https://doi.org/10.1016/j.jmr.2015.08.021">Symmetry constraints on spin dynamics: Application to hyperpolarized NMR</a>, Journal of Chemical Physics, 102:8 (2016).
%H A362534 Ole W. Sørensen, <a href="https://doi.org/10.1016/0022-2364(90)90278-H">A universal bound on spin dynamics</a>, Journal of Magnetic Resonance, 262 (1990) (appendix gives proof of the expression for symmetry constrained bounds).
%F A362534 a(n) = numerator(ceiling(n/2)/(2^(n)*binomial(n,ceiling(n/2))^(-1) - 1)).
%t A362534 Table[Numerator[Ceiling[n/2]  (2^n Binomial[n, Ceiling[n/2]]^-1 - 1 )^-1], {n, 1, 20}]
%o A362534 (PARI) a(n) = numerator(ceil(n/2)/(2^(n)*binomial(n,ceil(n/2))^(-1) - 1)); \\ _Michel Marcus_, Apr 25 2023
%Y A362534 Cf. A001803, A086116, A120778, A141244 (denominators but shifted).
%K A362534 nonn,frac
%O A362534 1,3
%A A362534 _Mohamed Sabba_, Apr 24 2023