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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).
5
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, 16, 1, 4, -6, -32, 1, 64, 20, -20, 1, 4, -6, -44, -19, 140, 136, -120, -132, 1, 5, -10, -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
OFFSET
0,8
COMMENTS
Triangle begins with T(0,0).
Two color patterns are equivalent if we permute the colors. Achiral color patterns must be equivalent if we reverse the order of the pattern.
The generating function for exactly n colors (column n of A304972) is
x^n * Sum_{k=0..n} (T(n, k) * x^k) / Product_{k=1..n} (1 - k*x^2).
Both the numerator and denominator of this g.f. have factors of (1+x) and (1-(n-2)*x^2) when n > 2.
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
[m==0 mod 2] * Sum_{k=0..n/2} T(n, 2k) * S2((m+n)/2-k, n) +
[m==1 mod 2] * Sum_{k=1..n/2} T(n, 2k-1) * S2((m+n+1)/2-k, n).
When n is odd, the function for A304972(m,n) is
[m==0 mod 2] * Sum_{k=0..(n-1)/2} T(n, 2k+1) * S2((m+n-1)-k, n) +
[m==1 mod 2] * Sum_{k=0..(n-1)/2} T(n, 2k) * S2((m+n)/2-k, n).
FORMULA
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].
EXAMPLE
Triangle begins:
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, 16;
1, 4, -6, -32, 1, 64, 20, -20;
1, 4, -6, -44, -19, 140, 136, -120, -132;
1, 5, -10, -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, -3028, 2236, 3600, 936;
MATHEMATICA
Coef[n_, -1] := Coef[n, -1] = 0; Coef[n_, 0] := Coef[n, 0] = Boole[n>=0];
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]]
Table[Coef[n, k], {n, 0, 30}, {k, 0, n}] // Flatten
CROSSREFS
Coefficients for functions and generating functions of A304973, A304974, A304975, A304976, which are columns 3-6 of A304972.
Sequence in context: A316784 A284974 A293222 * A245037 A161311 A161245
KEYWORD
sign,tabl,easy
AUTHOR
Robert A. Russell, May 23 2018
STATUS
approved