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%I A305008 #22 Jun 28 2018 03:15:51
%S A305008 1,1,1,1,1,0,1,2,-1,-2,1,2,-1,-4,-2,1,3,-3,-11,0,6,1,3,-3,-17,-8,20,
%T A305008 16,1,4,-6,-32,1,64,20,-20,1,4,-6,-44,-19,140,136,-120,-132,1,5,-10,
%U A305008 -70,5,301,152,-396,-280,28,1,5,-10,-90,-35,541,608,-1228,-1752,800,1216,1,6,-15,-130,15,966,643,-2798
%N A305008 Triangle read by rows of coefficients for functions and generating functions for the number of achiral color patterns (set partitions) for a row or loop of varying length using exactly n colors (sets).
%C A305008 Triangle begins with T(0,0).
%C A305008 Two color patterns are equivalent if we permute the colors. Achiral color patterns must be equivalent if we reverse the order of the pattern.
%C A305008 The generating function for exactly n colors (column n of A304972) is
%C A305008   x^n * Sum_{k=0..n} (T(n, k) * x^k) / Product_{k=1..n} (1 - k*x^2).
%C A305008 Both the numerator and denominator of this g.f. have factors of (1+x) and (1-(n-2)*x^2) when n > 2.
%C A305008 Letting S2(m,n) be the Stirling subset number A008277(m,n), the function for exactly n colors for a row or loop of length m, A304972(m,n), n even, is
%C A305008   [m==0 mod 2] * Sum_{k=0..n/2} T(n, 2k) * S2((m+n)/2-k, n) +
%C A305008   [m==1 mod 2] * Sum_{k=1..n/2} T(n, 2k-1) * S2((m+n+1)/2-k, n).
%C A305008 When n is odd, the function for A304972(m,n) is
%C A305008   [m==0 mod 2] * Sum_{k=0..(n-1)/2} T(n, 2k+1) * S2((m+n-1)-k, n) +
%C A305008   [m==1 mod 2] * Sum_{k=0..(n-1)/2} T(n, 2k) * S2((m+n)/2-k, n).
%F A305008 T(n,k) = [1 <= k <= n] * (T(n-1, k-1) + T(n-2, k) - (n-1) * T(n-2, k-2)) + [k==0 & n>=0].
%e A305008 Triangle begins:
%e A305008 1;
%e A305008 1, 1;
%e A305008 1, 1,   0;
%e A305008 1, 2,  -1,   -2;
%e A305008 1, 2,  -1,   -4,  -2;
%e A305008 1, 3,  -3,  -11,   0,   6;
%e A305008 1, 3,  -3,  -17,  -8,  20,  16;
%e A305008 1, 4,  -6,  -32,   1,  64,  20,   -20;
%e A305008 1, 4,  -6,  -44, -19, 140, 136,  -120,  -132;
%e A305008 1, 5, -10,  -70,   5, 301, 152,  -396,  -280,   28;
%e A305008 1, 5, -10,  -90, -35, 541, 608, -1228, -1752,  800, 1216;
%e A305008 1, 6, -15, -130,  15, 966, 643, -2798, -3028, 2236, 3600, 936;
%t A305008 Coef[n_, -1] := Coef[n, -1] = 0; Coef[n_, 0] := Coef[n, 0] = Boole[n>=0];
%t A305008 Coef[n_, k_] := Coef[n, k] = If[k > n, 0, Coef[n-1, k-1] + Coef[n-2, k] - (n-1) Coef[n-2, k-2]]
%t A305008 Table[Coef[n, k], {n, 0, 30}, {k, 0, n}] // Flatten
%Y A305008 Coefficients for functions and generating functions of A304973, A304974, A304975, A304976, which are columns 3-6 of A304972.
%K A305008 sign,tabl,easy
%O A305008 0,8
%A A305008 _Robert A. Russell_, May 23 2018

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