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

0, 0, 1, 7, 36, 168, 736, 3151, 13190, 54938, 226597, 934200, 3831219, 15723801, 64313623, 263316219, 1075420890, 4396310382, 17937457304, 73247306563, 298635873550, 1218428664338

Offset

1,4

Comments

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

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

Links

Table of n, a(n) for n=1..22.

M. E. Fisher and B. J. Hiley, Configuration and free energy of a polymer molecule with solvent interaction, J. Chem. Phys., 34 (1961), 1253-1267.

A. M. Nemirovsky, K. F. Freed, T. Ishinabe, and J. F. Douglas, Marriage of exact enumeration and 1/d expansion methods: lattice model of dilute polymers, J. Statist. Phys., 67 (1992), 1083-1108; see Eq. 5 (p. 1090).

Example

From Bert Dobbelaere, Jan 03 2019: (Start)
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:
a(3) = 1: { XYZ }
a(4) = 7: { XXYZ, XYXZ, XYYZ, XYZX, XYZx, XYZY, XYZZ }
a(5) = 36: {
      XXXYZ, XXYXZ, XXYYZ, XXYZX, XXYZx, XXYZY, XXYZZ, XYXXZ, XYXYZ,
      XYXZX, XYXZY, XYXZy, XYXZZ, XYYXZ, XYYxZ, XYYYZ, XYYZX, XYYZx,
      XYYZY, XYYZZ, XYZXX, XYZXY, XYZXy, XYZXZ, XYZxx, XYZxY, XYZxZ,
      XYZYX, XYZYx, XYZYY, XYZYZ, XYZZX, XYZZx, XYZZY, XYZZy, XYZZZ }
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.
(End)

Crossrefs

Cf. A002934, A174319, A323063.

Sequence in context: A095931 A292486 A026856 * A099455 A102053 A058681

Adjacent sequences: A038745 A038746 A038747 * A038749 A038750 A038751

Keyword

nonn,more

Author

N. J. A. Sloane, May 02 2000

Extensions

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.

a(12)-a(15) confirmed by direct computation and a(16)-a(22) from Bert Dobbelaere, Jan 03 2019

Status

approved