|
%I #26 Sep 08 2022 08:46:21
%S 0,0,0,0,1,2,9,16,53,90,265,440,1221,2002,5369,8736,22933,37130,96105,
%T 155080,397541,640002,1629529,2619056,6636213,10653370,26899145,
%U 43144920,108659461,174174002,437826489,701478976,1760871893,2820264810,7072185385,11324105960,28374834981,45425564002,113757620249
%N Number of achiral color patterns (set partitions) for a row or loop of length n using exactly 4 colors (sets).
%C Two color patterns are equivalent if we permute the colors. Achiral color patterns must be equivalent if we reverse the order of the pattern.
%H G. C. Greubel, <a href="/A304974/b304974.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,7,-7,-12,12).
%F a(n) = [n==0 mod 2] * (S2(n/2+2, 4) - S2(n/2+1, 4) - 2*S2(n/2, 4)) + [n==1 mod 2] * (2*S2((n+3)/2, 4) - 4*S2((n+1)/2, 4)) where S2(n,k) is the Stirling subset number A008277(n,k).
%F G.f.: x^4 * (1+x)^2 * (1-2x^2) / Product_{k=1..4} (1 - k*x^2).
%F a(n) = A304972(n,4).
%F a(2m-1) = A140735(m,4).
%F a(2m) = A293181(m,4).
%e For a(6) = 9, the row color patterns are AABCDD, ABACDC, ABBCCD, ABCADC, ABCBCD, ABCCBD, ABCCDA, ABCDAB, and ABCBCD. The loop color patterns are AAABCD, AABBCD, AABCCD, AABCDB, ABABCD, ABACAD, ABACBD, ABACDC, and ABCADC.
%t Table[If[EvenQ[n], StirlingS2[n/2 + 2, 4] - StirlingS2[n/2 + 1, 4] - 2 StirlingS2[n/2, 4], 2 StirlingS2[(n + 3)/2, 4] - 4 StirlingS2[(n + 1)/2, 4]], {n, 0, 40}]
%t Join[{0}, LinearRecurrence[{1, 7, -7, -12, 12}, {0, 0, 0, 1, 2}, 40]] (* _Robert A. Russell_, Oct 14 2018 *)
%o (PARI) m=40; v=concat([0,0,0,1,2], vector(m-5)); for(n=6, m, v[n] = v[n-1] +7*v[n-2] -7*v[n-3] -12*v[n-4] +12*v[n-5]); concat([0], v) \\ _G. C. Greubel_, Oct 17 2018
%o (Magma) I:=[0,0,0,1,2]; [0] cat [n le 5 select I[n] else Self(n-1) +7*Self(n-2) -7*Self(n-3) -12*Self(n-4) +12*Self(n-5): n in [1..40]]; // _G. C. Greubel_, Oct 17 2018
%Y Fourth column of A304972.
%Y Fourth column of A140735 for odd n.
%Y Fourth column of A293181 for even n.
%Y Coefficients that determine the first formula and generating function are row 4 of A305008.
%K nonn,easy
%O 0,6
%A _Robert A. Russell_, May 22 2018
|
