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A321391
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Array read by antidiagonals: T(n,k) is the number of achiral rows of n colors using up to k colors.
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7
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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, 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, 81, 16, 1, 0, 1, 10, 9, 64, 49, 216, 125, 256, 81, 32, 1, 0
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OFFSET
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0,8
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COMMENTS
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The antidiagonals go from top-right to bottom-left.
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LINKS
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FORMULA
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T(n,k) = [n==0] + [n>0] * k^ceiling(n/2).
The generating function for column k is (1+k*x) / (1-k*x^2).
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EXAMPLE
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The array begins with T(0,0):
1 1 1 1 1 1 1 1 1 1 1 1 ...
0 1 2 3 4 5 6 7 8 9 10 11 ...
0 1 2 3 4 5 6 7 8 9 10 11 ...
0 1 4 9 16 25 36 49 64 81 100 121 ...
0 1 4 9 16 25 36 49 64 81 100 121 ...
0 1 8 27 64 125 216 343 512 729 1000 1331 ...
0 1 8 27 64 125 216 343 512 729 1000 1331 ...
0 1 16 81 256 625 1296 2401 4096 6561 10000 14641 ...
0 1 16 81 256 625 1296 2401 4096 6561 10000 14641 ...
0 1 32 243 1024 3125 7776 16807 32768 59049 100000 161051 ...
0 1 32 243 1024 3125 7776 16807 32768 59049 100000 161051 ...
0 1 64 729 4096 15625 46656 117649 262144 531441 1000000 1771561 ...
For T(3,3)=9, the rows are AAA, ABA, ACA, BAB, BBB, BCB, CAC, CBC, and CCC.
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MATHEMATICA
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Table[If[n>0, (n-k)^Ceiling[k/2], 1], {n, 0, 12}, {k, 0, n}] // Flatten
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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