A086116 Numerator of mean deviation of a symmetrical binomial distribution on n elements.

1, 1, 3, 3, 15, 15, 35, 35, 315, 315, 693, 693, 3003, 3003, 6435, 6435, 109395, 109395, 230945, 230945, 969969, 969969, 2028117, 2028117, 16900975, 16900975, 35102025, 35102025, 145422675, 145422675, 300540195, 300540195, 9917826435

Offset: 1

Author

Eric W. Weisstein, Jul 10 2003

Keywords

• nonn
• frac

Comments

From Mohamed Sabba, Apr 24 2023: (Start)

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:

(a) in AX3 and AX4 spin systems, the maximum achievable transfer from Xz to Az is 3/2 = (2*3/4).

(b) in AX5 and AX6 spin systems, the maximum achievable transfer from Xz to Az is 15/8 = (2*15/16).

Note that this is different from the related adiabatic polarization transfer bounds, given by A141244.

(End)

Links

• Ole W. Sørensen, A universal bound on spin dynamics, Journal of Magnetic Resonance, 86:2 (1990) (equations 20 and 21)
• Ole W. Sørensen, Polarization transfer experiments in high-resolution NMR spectroscopy, Progress in Nuclear Magnetic Resonance Spectroscopy, 21:6 (1989), (page 534)
• Eric Weisstein's World of Mathematics, Random Walk 1-Dimensional
• Eric Weisstein's World of Mathematics, Binomial Distribution

Crossrefs

Cf. A001803, A086117, A141244.

Programs

Mathematica

Numerator[Table[If[OddQ[n], n!!/2/(n-1)!!, (n-1)!!/2/(n-2)!! ], {n, 50}]]