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A305625 Number of chiral pairs of rows of n colors with exactly 5 different colors.

%I A305625 #16 Sep 27 2018 03:13:30
%S A305625 0,0,0,0,60,900,8400,63000,417000,2551440,14802900,82763100,450501660,
%T A305625 2404493700,12645952200,65771370000,339164682000,1737485315640,
%U A305625 8855354531100,44952362878500,227475739300260,1148269299919500,5785013208282000,29100046926951000,146201097996135000,733811769167043840,3680292427100043300,18446421887430345900,92412024657725026860,462780012983867889300,2316780309783100387800
%N A305625 Number of chiral pairs of rows of n colors with exactly 5 different colors.
%C A305625 If the row is achiral, i.e., the same as its reverse, we ignore it. If different from its reverse, we count it and its reverse as a chiral pair.
%F A305625 a(n) = (k!/2) * (S2(n,k) - S2(ceiling(n/2),k)), with k=5 colors used and where S2(n,k) is the Stirling subset number A008277.
%F A305625 a(n) = (A001118(n) - A056456(n)) / 2.
%F A305625 a(n) = A001118(n) - A056312(n) = A056312(n) - A056456(n).
%F A305625 G.f.: -(k!/2) * (x^(2k-1) + x^(2k)) / Product_{j=1..k} (1 - j*x^2) + (k!/2) * x^k / Product_{j=1..k} (1 - j*x) with k=5 colors used.
%e A305625 For a(5) = 60, the chiral pairs are the 5! = 120 permutations of ABCDE, each paired with its reverse.
%t A305625 k=5; Table[(k!/2) (StirlingS2[n,k] - StirlingS2[Ceiling[n/2],k]), {n, 1, 40}]
%o A305625 (PARI) a(n) = 60*(stirling(n, 5, 2) - stirling(ceil(n/2), 5, 2)); \\ _Altug Alkan_, Sep 26 2018
%Y A305625 Fifth column of A305622.
%Y A305625 A056456(n) is number of achiral rows of n colors with exactly 5 different colors.
%Y A305625 Cf. A001118, A056312.
%K A305625 nonn,easy
%O A305625 1,5
%A A305625 _Robert A. Russell_, Jun 06 2018