A304974 Number of achiral color patterns (set partitions) for a row or loop of length n using exactly 4 colors (sets).

0, 0, 0, 0, 1, 2, 9, 16, 53, 90, 265, 440, 1221, 2002, 5369, 8736, 22933, 37130, 96105, 155080, 397541, 640002, 1629529, 2619056, 6636213, 10653370, 26899145, 43144920, 108659461, 174174002, 437826489, 701478976, 1760871893, 2820264810, 7072185385, 11324105960, 28374834981, 45425564002, 113757620249

Offset

0,6

Comments

Two color patterns are equivalent if we permute the colors. Achiral color patterns must be equivalent if we reverse the order of the pattern.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (1,7,-7,-12,12).

Formula

a(n) = [n==0 mod 2] * (S2(n/2+2, 4) - S2(n/2+1, 4) - 2*S2(n/2, 4)) + [n==1 mod 2] * (2*S2((n+3)/2, 4) - 4*S2((n+1)/2, 4)) where S2(n,k) is the Stirling subset number A008277(n,k).

G.f.: x^4 * (1+x)^2 * (1-2x^2) / Product_{k=1..4} (1 - k*x^2).

a(n) = A304972(n,4).

a(2m-1) = A140735(m,4).

a(2m) = A293181(m,4).

Example

For a(6) = 9, the row color patterns are AABCDD, ABACDC, ABBCCD, ABCADC, ABCBCD, ABCCBD, ABCCDA, ABCDAB, and ABCBCD.  The loop color patterns are AAABCD, AABBCD, AABCCD, AABCDB, ABABCD, ABACAD, ABACBD, ABACDC, and ABCADC.

Mathematica

Table[If[EvenQ[n], StirlingS2[n/2 + 2, 4] - StirlingS2[n/2 + 1, 4] - 2 StirlingS2[n/2, 4], 2 StirlingS2[(n + 3)/2, 4] - 4 StirlingS2[(n + 1)/2, 4]], {n, 0, 40}]
Join[{0}, LinearRecurrence[{1, 7, -7, -12, 12}, {0, 0, 0, 1, 2}, 40]] (* Robert A. Russell, Oct 14 2018 *)

Prog

(PARI) m=40; v=concat([0, 0, 0, 1, 2], vector(m-5)); for(n=6, m, v[n] = v[n-1] +7*v[n-2] -7*v[n-3] -12*v[n-4] +12*v[n-5]); concat([0], v) \\ G. C. Greubel, Oct 17 2018
(Magma) I:=[0, 0, 0, 1, 2]; [0] cat [n le 5 select I[n] else Self(n-1) +7*Self(n-2) -7*Self(n-3) -12*Self(n-4) +12*Self(n-5): n in [1..40]]; // G. C. Greubel, Oct 17 2018

Crossrefs

Fourth column of A304972.

Fourth column of A140735 for odd n.

Fourth column of A293181 for even n.

Coefficients that determine the first formula and generating function are row 4 of A305008.

Sequence in context: A237282 A178440 A097965 * A075645 A185252 A055259

Adjacent sequences: A304971 A304972 A304973 * A304975 A304976 A304977

Keyword

nonn,easy

Author

Robert A. Russell, May 22 2018

Status

approved