A304976 Number of achiral color patterns (set partitions) for a row or loop of length n using exactly 6 colors (sets).

0, 0, 0, 0, 0, 0, 1, 3, 18, 46, 195, 461, 1696, 3836, 13097, 28819, 94094, 203322, 644911, 1376217, 4279692, 9051592, 27755013, 58319855, 176992090, 370087718, 1114496747, 2321721493, 6950406008, 14437363668, 43021681249, 89162536011, 264732674406, 547676535634

Offset

0,8

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 (1,16,-16,-91,91,216,-216,-180, 180).

Formula

a(n) = [n==0 mod 2] * (S2(n/2+3, 6) - 3*S2(n/2+2, 6) - 8*S2(n/2+1, 6) + 16*S2(n/2, 6)) + [n==1 mod 2] * (3*S2((n+5)/2, 6) - 17*S2((n+3)/2, 6) + 20*S2((n+1)/2, 6 )) where S2(n,k) is the Stirling subset number A008277(n,k).

G.f.: x^6 *(1+x)*(1-4*x^2)*(1+2*x-x^2-4*x^3) / Product_{k=1..6} (1 - k*x^2).

a(n) = A304972(n,6).

a(2m-1) = A140735(m,6).

a(2m) = A293181(m,6).

Example

For a(7) = 3, the color patterns for both rows and loops are ABCDCEF, ABCDEBF, and ABCDEFA.

Mathematica

Table[If[EvenQ[n], StirlingS2[n/2 + 3, 6] - 3 StirlingS2[n/2 + 2, 6] - 8 StirlingS2[n/2 + 1, 6] + 16 StirlingS2[n/2, 6], 3 StirlingS2[(n + 5)/2, 6] - 17 StirlingS2[(n + 3)/2, 6] + 20 StirlingS2[(n + 1)/2, 6]], {n, 0, 40}]
Join[{0}, LinearRecurrence[{1, 16, -16, -91, 91, 216, -216, -180, 180}, {0, 0, 0, 0, 0, 1, 3, 18, 46}, 40]] (* Robert A. Russell, Oct 14 2018 *)
CoefficientList[Series[x^6 *(1+x)*(1-4*x^2)*(1+2*x-x^2-4*x^3) / Product[1 - k*x^2, {k, 1, 6}], {x, 0, 50}], x] (* Stefano Spezia, Oct 20 2018 *)

Prog

(PARI) m=40; v=concat([0, 0, 0, 0, 0, 1, 3, 18, 46], vector(m-9)); for(n=10, m, v[n] = v[n-1] +16*v[n-2] -16*v[n-3] -91*v[n-4] +91*v[n-5] +216*v[n-6] -216*v[n-7] -180*v[n-8] +180*v[n-9]); concat([0], v) \\ G. C. Greubel, Oct 16 2018
(Magma) I:=[0, 0, 0, 0, 0, 1, 3, 18, 46]; [0] cat [n le 9 select I[n] else Self(n-1) +16*Self(n-2) -16*Self(n-3) -91*Self(n-4) +91*Self(n-5) +216*Self(n-6) -216*Self(n-7) -180*Self(n-8) +180*Self(n-9): n in [1..40]]; // G. C. Greubel, Oct 16 2018

Crossrefs

Sixth column of A304972.

Sixth column of A140735 for odd n.

Sixth column of A293181 for even n.

Coefficients that determine the first formula and generating function are row 6 of A305008.

Sequence in context: A094159 A138976 A275038 * A364599 A064043 A267639

Adjacent sequences: A304973 A304974 A304975 * A304977 A304978 A304979

Keyword

nonn,easy

Author

Robert A. Russell, May 22 2018

Status

approved