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

Author

Robert A. Russell, May 22 2018

Keywords

• nonn
• easy

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.

Examples

			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.
		

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).

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.

Programs

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

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 *)

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

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).