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Search: id:A103323
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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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+0
8
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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
(list; table; graph; listen)
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OFFSET
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1,6
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COMMENT
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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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FORMULA
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T(n, k) = A000045(k)^n, n, k > 0.
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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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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PROGRAM
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(PARI) T(n, k)=fibonacci(k)^n
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CROSSREFS
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Rows include A000045, A007598, A056570, A056571, A056572, A056573, A056574.
Adjacent sequences: A103320 A103321 A103322 this_sequence A103324 A103325 A103326
Sequence in context: A099239 A009998 A113993 this_sequence A092056 A103574 A112682
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KEYWORD
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nonn,tabl
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AUTHOR
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Ralf Stephan, Feb 02 2005
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