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A038748 Coefficients arising in the enumeration of configurations of linear chains.

%I A038748 #37 Feb 24 2019 02:05:56
%S A038748 0,0,1,7,36,168,736,3151,13190,54938,226597,934200,3831219,15723801,
%T A038748 64313623,263316219,1075420890,4396310382,17937457304,73247306563,
%U A038748 298635873550,1218428664338
%N A038748 Coefficients arising in the enumeration of configurations of linear chains.
%C A038748 In the notation of Nemirovsky et al. (1992), a(n), the n-th term of the current sequence, is equal to p_{n,m}^{(l)} with m = 0 and l = 3. - _Petros Hadjicostas_, Jan 02 2019
%C A038748 This counts non-self-intersecting paths of length n on the cubic lattice, start and end points distinguished, planar paths not counted, rotations and reflections of a path not counted as distinct from that path. No points repeated, no adjacent points allowed unless consecutive in path. - _Bert Dobbelaere_, Jan 03 2019
%H A038748 M. E. Fisher and B. J. Hiley, <a href="http://dx.doi.org/10.1063/1.1731729">Configuration and free energy of a polymer molecule with solvent interaction</a>, J. Chem. Phys., 34 (1961), 1253-1267.
%H A038748 A. M. Nemirovsky, K. F. Freed, T. Ishinabe, and J. F. Douglas, <a href="http://dx.doi.org/10.1007/BF01049010">Marriage of exact enumeration and 1/d expansion methods: lattice model of dilute polymers</a>, J. Statist. Phys., 67 (1992), 1083-1108; see Eq. 5 (p. 1090).
%e A038748 From _Bert Dobbelaere_, Jan 03 2019: (Start)
%e A038748 Using strings to represent a path with characters X,Y,Z for steps in positive directions and x,y,z for steps in negative directions along the respective axes, the following enumerations correspond to the first nonzero terms:
%e A038748 a(3) = 1: { XYZ }
%e A038748 a(4) = 7: { XXYZ, XYXZ, XYYZ, XYZX, XYZx, XYZY, XYZZ }
%e A038748 a(5) = 36: {
%e A038748       XXXYZ, XXYXZ, XXYYZ, XXYZX, XXYZx, XXYZY, XXYZZ, XYXXZ, XYXYZ,
%e A038748       XYXZX, XYXZY, XYXZy, XYXZZ, XYYXZ, XYYxZ, XYYYZ, XYYZX, XYYZx,
%e A038748       XYYZY, XYYZZ, XYZXX, XYZXY, XYZXy, XYZXZ, XYZxx, XYZxY, XYZxZ,
%e A038748       XYZYX, XYZYx, XYZYY, XYZYZ, XYZZX, XYZZx, XYZZY, XYZZy, XYZZZ }
%e A038748 Symmetries are avoided by imposing the following restrictions: all patterns start with 'X'. First occurrence of 'Y' comes before the first occurrence of 'Z' (presence mandatory). First occurrence of steps in negative directions (presence optional) comes after the first occurrence of the corresponding steps in positive directions.
%e A038748 (End)
%Y A038748 Cf. A002934, A174319, A323063.
%K A038748 nonn,more
%O A038748 1,4
%A A038748 _N. J. A. Sloane_, May 02 2000
%E A038748 Terms a(12) to a(15) were calculated by _Petros Hadjicostas_, Jan 01 2019 using Eq. (5) in Nemirovsky et al. (1992) and the terms of the sequences A038746 and A174319.
%E A038748 a(12)-a(15) confirmed by direct computation and a(16)-a(22) from _Bert Dobbelaere_, Jan 03 2019