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A305751 Number of achiral color patterns (set partitions) in a row or cycle of length n with 5 or fewer colors (subsets). 5
1, 1, 2, 3, 7, 12, 30, 55, 141, 266, 688, 1313, 3407, 6532, 16970, 32595, 84721, 162846, 423348, 813973, 2116227, 4069352, 10580110, 20345735, 52898501, 101726626, 264488408, 508629033, 1322433847, 2543136972, 6612152850, 12715668475 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

An equivalent color pattern is obtained when we permute the colors. Thus all permutations of ABCDE are equivalent, as are AABCDE and BBCDEA. A color pattern is achiral if it is equivalent to its reversal. Rotations of the colors of a cycle are equivalent, so for cycles AABBCDE =  BBCDEAA = CDEAABB.

LINKS

Table of n, a(n) for n=0..31.

Index entries for linear recurrences with constant coefficients, signature (1,7,-7,-10,10).

FORMULA

a(n) = Sum_{j=0..5} Ach(n,j), where Ach(n,k) = [n>1] * (k*T(n-2,k) + T(n-2,k-1) + T(n-2,k-2)) + [0<=n<2 & n==k].

G.f.: (1 - 2x)*(1+2x-2x^2-3x^3+x^4) / ((1-x)*(1-2x^2)*(1-5x^2)).

a(2m) = S2(m+5,5) - 13*S2(m+4,5) + 62*S2(m+3,5) - 130*S2(m+2,5) + 110*S2(m+1,5) - 24*S2(m,5);

a(2m-1) = S2(m+4,5) - 12*S2(m+3,5) + 52*S2(m+2,5) - 95*S2(m+1,5) + 60*S2(m,5), where S2(n,k) is the Stirling subset number A008277.

For n>0, a(2m) = (15 + 20*2^m + 13*5^m) / 60.

a(2m-1) = (3 + 2*2^m + 5^m) / 12.

a(n) = 2*A056324(n) - A056272(n) = A056272(n) - 2*A320935(n) = A056324(n) - A320935(n).

a(n) = 2*A056355(n) - A056293(n) = A056293(n) - 2*A320745(n) = A056355(n) - A320745(n).

a(n) = A057427(n) + A052551(n-2) + A304973(n) + A304974(n) + A304975(n).

a(n) = a(n-1) + 7*a(n-2) - 7*a(n-3) - 10*a(n-4) + 10*a(n-5). - Muniru A Asiru, Oct 30 2018

EXAMPLE

For a(5) = 12, the achiral patterns for both rows and cycles are AAAAA, AABAA, ABABA, ABBBA, AABCC, ABACA, ABBBC, ABCAB, ABCBA, ABCBD, ABCDA, and ABCDE.

MAPLE

seq(coeff(series((1-2*x)*(1+2*x-2*x^2-3*x^3+x^4)/((1-x)*(1-2*x^2)*(1-5*x^2)), x, n+1), x, n), n = 0 .. 40); # Muniru A Asiru, Oct 30 2018

MATHEMATICA

Table[If[EvenQ[n], StirlingS2[(n+10)/2, 5] - 13 StirlingS2[(n+8)/2, 5] + 62 StirlingS2[(n+6)/2, 5] - 130 StirlingS2[(n+4)/2, 5] + 110 StirlingS2[(n+2)/2, 5] - 24 StirlingS2[n/2, 5], StirlingS2[(n+9)/2, 5] - 12 StirlingS2[(n+7)/2, 5] + 52 StirlingS2[(n+5)/2, 5] - 95 StirlingS2[(n+3)/2, 5] + 60 StirlingS2[(n+1)/2, 5]], {n, 0, 40}]

Ach[n_, k_] := Ach[n, k] = If[n<2, Boole[n==k && n>=0], k Ach[n-2, k] + Ach[n-2, k-1] + Ach[n-2, k-2]]; (* A304972 *)

k=5; Table[Sum[Ach[n, j], {j, 0, k}], {n, 0, 40}]

CoefficientList[Series[(1-2x)(1+2x-2x^2-3x^3+x^4) / ((1- x)(1-2x^2)(1-5x^2)), {x, 0, 40}], x]

Join[{1}, LinearRecurrence[{1, 7, -7, -10, 10}, {1, 2, 3, 7, 12}, 40]]

Join[{1}, Table[If[EvenQ[n], (15 + 20 2^(n/2) + 13 5^(n/2)) / 60, (3 + 2 2^((n+1)/2) + 5^((n+1)/2)) / 12], {n, 40}]]

CROSSREFS

Fifth column of A305749.

Cf. A056272 (oriented), A056324 (unoriented), A320935 (chiral), for rows.

Cf. A056293 (oriented), A056355 (unoriented), A320745 (chiral), for cycles.

Sequence in context: A339159 A297438 A111759 * A047749 A134565 A300749

Adjacent sequences:  A305748 A305749 A305750 * A305752 A305753 A305754

KEYWORD

nonn,easy

AUTHOR

Robert A. Russell, Jun 09 2018

STATUS

approved

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Last modified October 1 00:23 EDT 2022. Contains 357114 sequences. (Running on oeis4.)