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A103323
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Square array T(n,k) read by antidiagonals: powers of Fibonacci numbers.
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13
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1, 1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 8, 9, 5, 1, 1, 16, 27, 25, 8, 1, 1, 32, 81, 125, 64, 13, 1, 1, 64, 243, 625, 512, 169, 21, 1, 1, 128, 729, 3125, 4096, 2197, 441, 34, 1, 1, 256, 2187, 15625, 32768, 28561, 9261, 1156, 55, 1, 1, 512, 6561, 78125, 262144, 371293, 194481, 39304, 3025, 89
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OFFSET
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1,6
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COMMENTS
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Number of ways to create subsets S(1), S(2),..., S(k-1) such that S(1) is in [n] and for 2<=i<=k-1, S(i) is in [n] and S(i) is disjoint from S(i-1).
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REFERENCES
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A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, identity 138.
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LINKS
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FORMULA
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T(n, k) = Sum[i_1>=0, Sum[i_2>=0, ... Sum[i_{k-1}>=0, C(n, i_1)*C(n-i_1, i_2)*C(n-i_2, i_3)*...*C(n-i_{k-2}, i_{k-1}) ] ... ]].
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EXAMPLE
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Square array T(n,k) begins:
1, 1, 2, 3, 5, 8, ...
1, 1, 4, 9, 25, 64, ...
1, 1, 8, 27, 125, 512, ...
1, 1, 16, 81, 625, 4096, ...
1, 1, 32, 243, 3125, 32768, ...
1, 1, 64, 729, 15625, 262144, ...
...
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MAPLE
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A:= (n, k)-> (<<1|1>, <1|0>>^n)[1, 2]^k:
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MATHEMATICA
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T[n_, k_] := Fibonacci[k]^n; Table[T[n-k+1, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 16 2015 *)
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PROG
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(PARI) T(n, k)=fibonacci(k)^n
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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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