A304975 Number of achiral color patterns (set partitions) for a row or loop of length n using exactly 5 colors (sets).

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.

Links

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

Index entries for linear recurrences with constant coefficients, signature (1,11,-11,-38,38,40,-40).

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

Adjacent sequences: A304972 A304973 A304974 * A304976 A304977 A304978

Keyword

nonn,easy

Author

Robert A. Russell, May 22 2018

Status

approved