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A304976
Number of achiral color patterns (set partitions) for a row or loop of length n using exactly 6 colors (sets).
7
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
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
KEYWORD
nonn,easy
AUTHOR
Robert A. Russell, May 22 2018
STATUS
approved