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 A140735 Triangle read by rows, X^n * [1,0,0,0,...]; where X = a tridiagonal matrix with (1,2,3,...) in the main diagonal and (1,1,1,...) in the sub and subsubdiagonals. 11
 1, 1, 1, 1, 1, 3, 5, 2, 1, 1, 7, 19, 16, 12, 3, 1, 1, 15, 65, 90, 95, 46, 22, 4, 1, 1, 31, 211, 440, 630, 461, 295, 100, 35, 5, 1, 1, 63, 665, 2002, 3801, 3836, 3156, 1556, 710, 185, 51, 6, 1, 1, 127, 2059, 8736, 21672, 28819, 29729, 19440, 11102, 4116, 1456, 308, 70 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,6 COMMENTS T(m,k) is the number of achiral color patterns in a row or loop of length 2m-1 using exactly k different colors. Two color patterns are equivalent if we permute the colors. - Robert A. Russell, Mar 24 2018 LINKS FORMULA G.f.(exponential in x, ordinary in t): exp(x+t*(exp(x)-1)+(1/2)*t^2*(exp(2*x)-1)). - Ira M. Gessel, Jan 30 2018 T(m,k) = [m>1]*(k*T(m-1,k)+T(m-1,k-1)+T(m-1,k-2)) + [m==1]*[k==1] - Robert A. Russell, Apr 24 2018 EXAMPLE First few rows of the triangle are:   1;   1,  1,   1;   1,  3,   5,    2,    1;   1,  7,  19,   16,   12,    3,    1;   1, 15,  65,   90,   95,   46,   22,    4,   1;   1, 31, 211,  440,  630,  461,  295,  100,  35,   5,  1;   1, 63, 665, 2002, 3801, 3836, 3156, 1556, 710, 185, 51, 6, 1;   ... T(3,3)=5 is the number of achiral color patterns of length five using exactly three colors. These are AABCC, ABACA, ABBBC, ABCAB, and ABCBA for both rows and loops. - Robert A. Russell, Mar 24 2018 MATHEMATICA (* Ach[n, k] is the number of achiral color patterns for a row or loop of n   colors containing k different colors *) 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, 13, 2}, {k, 1, n}] // Flatten (* Robert A. Russell, Feb 06 2018 *) Table[MatrixPower[Table[Switch[j-i, 0, i, 1, 1, 2, 1, _, 0],   {i, 1, 2 n - 1}, {j, 1, 2 n - 1}], n-1][[1]], {n, 1, 10}]   // Flatten (* Robert A. Russell, Mar 24 2018 *) Aodd[m_, k_] := Aodd[m, k] = If[m > 1, k Aodd[m-1, k] + Aodd[m-1, k-1]   + Aodd[m-1, k-2], Boole[m==1 && k==1]] Table[Aodd[m, k], {m, 1, 10}, {k, 1, 2m-1}] // Flatten (* Robert A. Russell, Apr 24 2018 *) CROSSREFS Cf. A080337 (row sums), A140733, A140744. Number of achiral color patterns of length even n in A293181. Sequence in context: A220110 A240752 A021288 * A183206 A197521 A161865 Adjacent sequences:  A140732 A140733 A140734 * A140736 A140737 A140738 KEYWORD nonn,tabf AUTHOR Gary W. Adamson, May 25 2008 STATUS approved

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Last modified September 22 16:56 EDT 2019. Contains 327311 sequences. (Running on oeis4.)