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%I #18 Jun 11 2023 00:06:07
%S 1,1,3,3,15,15,35,35,315,315,693,693,3003,3003,6435,6435,109395,
%T 109395,230945,230945,969969,969969,2028117,2028117,16900975,16900975,
%U 35102025,35102025,145422675,145422675,300540195,300540195,9917826435
%N Numerator of mean deviation of a symmetrical binomial distribution on n elements.
%C From _Mohamed Sabba_, Apr 24 2023: (Start)
%C The numerators of this sequence and denominators (A086117) appear in NMR and spin physics as half of the symmetry-constrained upper bound on polarization transfer in AXn spin-1/2 systems. For example:
%C (a) in AX3 and AX4 spin systems, the maximum achievable transfer from Xz to Az is 3/2 = (2*3/4).
%C (b) in AX5 and AX6 spin systems, the maximum achievable transfer from Xz to Az is 15/8 = (2*15/16).
%C Note that this is different from the related adiabatic polarization transfer bounds, given by A141244.
%C (End)
%H 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, 86:2 (1990) (equations 20 and 21)
%H Ole W. Sørensen, <a href="/A086116/a086116.jpg">Polarization transfer experiments in high-resolution NMR spectroscopy</a>, Progress in Nuclear Magnetic Resonance Spectroscopy, 21:6 (1989), (page 534)
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RandomWalk1-Dimensional.html">Random Walk 1-Dimensional</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BinomialDistribution.html">Binomial Distribution</a>
%t Numerator[Table[If[OddQ[n], n!!/2/(n-1)!!, (n-1)!!/2/(n-2)!! ], {n, 50}]]
%Y Cf. A001803, A086117, A141244.
%K nonn,frac
%O 1,3
%A _Eric W. Weisstein_, Jul 10 2003
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