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Search: id:A106512
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| A106512 |
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Array read by antidiagonals: a(n,k) = number of k-colorings of a circle of n nodes (n >= 1, k >= 1). |
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1
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| 0, 0, 0, 0, 2, 0, 0, 6, 0, 0, 0, 12, 6, 2, 0, 0, 20, 24, 18, 0, 0, 0, 30, 60, 84, 30, 2, 0, 0, 42, 120, 260, 240, 66, 0, 0, 0, 56, 210, 630, 1020, 732, 126, 2, 0, 0, 72, 336, 1302, 3120, 4100, 2184, 258, 0, 0, 0, 90, 504, 2408, 7770, 15630, 16380, 6564, 510, 2, 0, 12, 110
(list; table; graph; listen)
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OFFSET
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1,5
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COMMENT
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Note that we keep one edge in the circular graph even when there's only one node (so there are 0 colorings of one node with k colors).
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FORMULA
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a(n, k) = (k-1)^n + (-1)^n * (k-1).
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EXAMPLE
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a(4,3) = 18 because there are three choices for the first node's color (call it 1) and then two choices for the second node's color (call it 2) and then the remaining two nodes can be 12, 13, or 32. So in total there are 3*2*3 = 18 ways. a(3,4) = 4*3*2 = 24 because the three nodes must be three distinct colors.
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CROSSREFS
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Cf. A092297, A090860.
Sequence in context: A078112 A128711 A132710 this_sequence A094785 A035536 A098643
Adjacent sequences: A106509 A106510 A106511 this_sequence A106513 A106514 A106515
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KEYWORD
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nonn,tabl
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AUTHOR
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Joshua Zucker (joshua.zucker(AT)stanfordalumni.org), May 29 2005
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