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%I #6 Oct 21 2022 21:55:39
%S 0,0,12,108,480,1500,3780,8232,16128,29160,49500,79860,123552,184548,
%T 267540,378000,522240,707472,941868,1234620,1596000,2037420,2571492,
%U 3212088,3974400,4875000,5931900,7164612,8594208,10243380,12136500
%N Number of chiral pairs of rows of length 5 using up to n colors.
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6, -15, 20, -15, 6, -1).
%F a(n) = (n^5 - n^3) / 2.
%F a(n) = (A000584(n) - A000578(n)) / 2.
%F a(n) = A000584(n) - A168178(n) = A168178(n) - A000578(n).
%F G.f.: (Sum_{j=1..5} S2(5,j)*j!*x^j/(1-x)^(j+1) - Sum_{j=1..3} S2(3,j)*j!*x^j/(1-x)^(j+1)) / 2, where S2 is the Stirling subset number A008277.
%F G.f.: x * Sum_{k=1..4} A145883(5,k) * x^k / (1-x)^6.
%F E.g.f.: (Sum_{k=1..5} S2(5,k)*x^k - Sum_{k=1..3} S2(3,k)*x^k) * exp(x) / 2, where S2 is the Stirling subset number A008277.
%F For n>5, a(n) = Sum_{j=1..6} -binomial(j-7,j) * a(n-j).
%e For a(0)=0 and a(1)=0, there are no chiral rows using fewer than two colors. For a(2)=12, the chiral pairs are AAAAB-BAAAA, AAABA-ABAAA, AAABB-BBAAA, AABAB-BABAA, AABBA-ABBAA, AABBB-BBBAAA, ABAAB-BAABA, ABABB-BBABA, ABBAB-BABBA, ABBBB-BBBBA, BAABB-BBAAB, and BABBB-BBBAB.
%t Table[(n^5-n^3)/2,{n,0,40}]
%t LinearRecurrence[{6, -15, 20, -15, 6, -1}, {0, 0, 12, 108, 480, 1500}, 40]
%o (PARI) a(n)=(n^5-n^3)/2 \\ _Charles R Greathouse IV_, Oct 21 2022
%Y Row 5 of A293500.
%Y Cf. A000584 (oriented), A168178 (unoriented), A000578 (achiral).
%K nonn,easy
%O 0,3
%A _Robert A. Russell_, Nov 16 2018
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