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%I #8 Nov 10 2018 09:55:36
%S 1,1,0,1,1,0,1,2,1,0,1,3,2,1,0,1,4,3,4,1,0,1,5,4,9,4,1,0,1,6,5,16,9,8,
%T 1,0,1,7,6,25,16,27,8,1,0,1,8,7,36,25,64,27,16,1,0,1,9,8,49,36,125,64,
%U 81,16,1,0,1,10,9,64,49,216,125,256,81,32,1,0
%N Array read by antidiagonals: T(n,k) is the number of achiral rows of n colors using up to k colors.
%C The antidiagonals go from top-right to bottom-left.
%F T(n,k) = [n==0] + [n>0] * k^ceiling(n/2).
%F The generating function for column k is (1+k*x) / (1-k*x^2).
%e The array begins with T(0,0):
%e 1 1 1 1 1 1 1 1 1 1 1 1 ...
%e 0 1 2 3 4 5 6 7 8 9 10 11 ...
%e 0 1 2 3 4 5 6 7 8 9 10 11 ...
%e 0 1 4 9 16 25 36 49 64 81 100 121 ...
%e 0 1 4 9 16 25 36 49 64 81 100 121 ...
%e 0 1 8 27 64 125 216 343 512 729 1000 1331 ...
%e 0 1 8 27 64 125 216 343 512 729 1000 1331 ...
%e 0 1 16 81 256 625 1296 2401 4096 6561 10000 14641 ...
%e 0 1 16 81 256 625 1296 2401 4096 6561 10000 14641 ...
%e 0 1 32 243 1024 3125 7776 16807 32768 59049 100000 161051 ...
%e 0 1 32 243 1024 3125 7776 16807 32768 59049 100000 161051 ...
%e 0 1 64 729 4096 15625 46656 117649 262144 531441 1000000 1771561 ...
%e For T(3,3)=9, the rows are AAA, ABA, ACA, BAB, BBB, BCB, CAC, CBC, and CCC.
%t Table[If[n>0, (n-k)^Ceiling[k/2], 1], {n, 0, 12}, {k, 0, n}] // Flatten
%Y Columns 0-6 are A000007, A000012, A060546, A056449, A056450, A056451, A056452.
%Y Cf. A003992 (oriented), A277504 (unoriented), A293500 (chiral).
%K nonn,easy,tabl
%O 0,8
%A _Robert A. Russell_, Nov 08 2018
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