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 A304972 Triangle read by rows of achiral color patterns (set partitions) for a row or loop of length n. T(n,k) is the number using exactly k colors (sets). 47
 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 3, 5, 2, 1, 1, 7, 10, 9, 3, 1, 1, 7, 19, 16, 12, 3, 1, 1, 15, 38, 53, 34, 18, 4, 1, 1, 15, 65, 90, 95, 46, 22, 4, 1, 1, 31, 130, 265, 261, 195, 80, 30, 5, 1, 1, 31, 211, 440, 630, 461, 295, 100, 35, 5, 1, 1, 63, 422, 1221, 1700, 1696, 1016, 515, 155, 45, 6, 1, 1, 63, 665, 2002 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,8 COMMENTS Two color patterns are equivalent if we permute the colors. Achiral color patterns must be equivalent if we reverse the order of the pattern. LINKS Andrew Howroyd, Table of n, a(n) for n = 1..1275 Ira Gessel, What is the number of achiral color patterns for a row of n colors containing k different colors?, mathoverflow, Jan 30 2018. FORMULA T(n,k) = [n>1] * (k*T(n-2,k) + T(n-2,k-1) + T(n-2,k-2)) + [n<2 & n==k & n>=0]. T(2m-1,k) = A140735(m,k). T(2m,k) = A293181(m,k). T(n,k) = [k==0 & n==0] + [k==1 & n>0]   + [k>1 & n==1 mod 2] * Sum_{i=0..(n-1)/2} (C((n-1)/2, i) * T(n-1-2i, k-1))   + [k>1 & n==0 mod 2] * Sum_{i=0..(n-2)/2} (C((n-2)/2, i) * (T(n-2-2i, k-1)   + 2^i * T(n-2-2i, k-2))) where C(n,k) is a binomial coefficient. EXAMPLE Triangle begins: 1; 1,   1; 1,   1,    1; 1,   3,    2,    1; 1,   3,    5,    2,     1; 1,   7,   10,    9,     3,     1; 1,   7,   19,   16,    12,     3,     1; 1,  15,   38,   53,    34,    18,     4,    1; 1,  15,   65,   90,    95,    46,    22,    4,    1; 1,  31,  130,  265,   261,   195,    80,   30,    5,    1; 1,  31,  211,  440,   630,   461,   295,  100,   35,    5,   1; 1,  63,  422, 1221,  1700,  1696,  1016,  515,  155,   45,   6,  1 1,  63,  665, 2002,  3801,  3836,  3156, 1556,  710,  185,  51,  6, 1; 1, 127, 1330, 5369, 10143, 13097, 10508, 6832, 2926, 1120, 266, 63, 7, 1; For T(4,2)=3, the row patterns are AABB, ABAB, and ABBA.  The loop patterns are AAAB, AABB, and ABAB. For T(5,3)=5, the color patterns for both rows and loops are AABCC, ABACA, ABBBC, ABCAB, and ABCBA. MATHEMATICA Ach[n_, k_] := Ach[n, k] = If[n < 2, Boole[n == k && n >= 0],   k Ach[n - 2, k] + Ach[n - 2, k - 1] + Ach[n - 2, k - 2]] Table[Ach[n, k], {n, 1, 15}, {k, 1, n}] // Flatten Ach[n_, k_] := Ach[n, k] = Which[0==k, Boole[0==n], 1==k, Boole[n>0],   OddQ[n], Sum[Binomial[(n-1)/2, i] Ach[n-1-2i, k-1], {i, 0, (n-1)/2}],   True, Sum[Binomial[n/2-1, i] (Ach[n-2-2i, k-1]   + 2^i Ach[n-2-2i, k-2]), {i, 0, n/2-1}]] Table[Ach[n, k], {n, 1, 15}, {k, 1, n}] // Flatten PROG (PARI) Ach(n)={my(M=matrix(n, n, i, k, i>=k)); for(i=3, n, for(k=2, n, M[i, k]=k*M[i-2, k] + M[i-2, k-1] + if(k>2, M[i-2, k-2]))); M} { my(A=Ach(10)); for(n=1, #A, print(A[n, 1..n])) } \\ Andrew Howroyd, Sep 18 2019 CROSSREFS Columns 1-6 are A057427, A052551(n-2), A304973, A304974, A304975, A304976. A305008 has coefficients that determine the function and generating function for each column. Row sums are A080107. Sequence in context: A087284 A262311 A242950 * A152176 A152175 A321620 Adjacent sequences:  A304969 A304970 A304971 * A304973 A304974 A304975 KEYWORD nonn,tabl,easy AUTHOR Robert A. Russell, May 22 2018 STATUS approved

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Last modified July 24 14:51 EDT 2021. Contains 346273 sequences. (Running on oeis4.)