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A304975
Number of achiral color patterns (set partitions) for a row or loop of length n using exactly 5 colors (sets).
8
0, 0, 0, 0, 0, 1, 3, 12, 34, 95, 261, 630, 1700, 3801, 10143, 21672, 57414, 119155, 314121, 639210, 1679320, 3370301, 8832483, 17549532, 45907994, 90541815, 236526381, 463889790, 1210585740, 2364180001, 6164760423, 11999840592, 31271161774, 60714998075, 158145313041, 306438236370, 797884712960
OFFSET
0,7
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.
FORMULA
a(n) = [n==0 mod 2] * (3*S2(n/2+2, 5) - 11*S2(n/2+1, 5) + 6*S2(n/2, 5)) + [n==1 mod 2] * (S2((n+5)/2, 5) - 3*S2((n+3)/2, 5)) where S2(n,k) is the Stirling subset number A008277(n,k).
G.f.: x^5 *(1 + x)*(1 - 3*x^2)*(1 + 2*x - 2*x^2) / Product_{k=1..5} (1 - k*x^2).
a(n) = A304972(n,5).
a(2m-1) = A140735(m,5).
a(2m) = A293181(m,5).
EXAMPLE
For a(6) = 3, the color patterns for both rows and loops are ABCCDE, ABCDBE, and ABCDEA.
MATHEMATICA
Table[If[EvenQ[n], 3 StirlingS2[n/2+2, 5] - 11 StirlingS2[n/2+1, 5] + 6 StirlingS2[n/2, 5], StirlingS2[(n+5)/2, 5] - 3 StirlingS2[(n+3)/2, 5]], {n, 0, 40}]
Join[{0}, LinearRecurrence[{1, 11, -11, -38, 38, 40, -40}, {0, 0, 0, 0, 1, 3, 12}, 40]] (* Robert A. Russell, Oct 14 2018 *)
CoefficientList[Series[x^5 *(1 + x)*(1 - 3*x^2)*(1 + 2*x - 2*x^2) / Product[1 - k*x^2, {k, 1, 5}], {x, 0, 50}], x] (* Stefano Spezia, Oct 16 2018 *)
PROG
(PARI) m=40; v=concat([0, 0, 0, 0, 1, 3, 12], vector(m-7)); for(n=8, m, v[n] = v[n-1] +11*v[n-2] -11*v[n-3] -38*v[n-4] +38*v[n-5] +40*v[n-6] -40*v[n-7] ); concat([0], v) \\ G. C. Greubel, Oct 16 2018
(Magma) I:=[0, 0, 0, 0, 1, 3, 12]; [0] cat [n le 7 select I[n] else Self(n-1) +11*Self(n-2) -11*Self(n-3) -38*Self(n-4) +38*Self(n-5) +40*Self(n-6) -40*Self(n-7): n in [1..40]]; // G. C. Greubel, Oct 16 2018
CROSSREFS
Fifth column of A304972.
Fifth column of A140735 for odd n.
Fifth column of A293181 for even n.
Coefficients that determine the first formula and generating function are row 5 of A305008.
Sequence in context: A184705 A257890 A060298 * A226546 A073372 A305023
KEYWORD
nonn,easy
AUTHOR
Robert A. Russell, May 22 2018
STATUS
approved