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 A103323 Square array T(n,k) read by antidiagonals: powers of Fibonacci numbers. 12
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,6 COMMENTS 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). REFERENCES A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, identity 138. LINKS Alois P. Heinz, Antidiagonals n = 1..100, flattened FORMULA 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}) ] ... ]]. EXAMPLE 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, ... MAPLE A:= (n, k)-> (<<1|1>, <1|0>>^n)[1, 2]^k: seq(seq(A(n, 1+d-n), n=1..d), d=1..12);  # Alois P. Heinz, Jun 17 2014 MATHEMATICA 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 *) PROG (PARI) T(n, k)=fibonacci(k)^n CROSSREFS Rows include A000045, A007598, A056570, A056571, A056572, A056573, A056574. Main diagonal gives A100399. Cf. A244003. Cf. A105317, A254719. Sequence in context: A167630 A009998 A113993 * A092056 A103574 A112682 Adjacent sequences:  A103320 A103321 A103322 * A103324 A103325 A103326 KEYWORD nonn,tabl AUTHOR Ralf Stephan, Feb 02 2005 STATUS approved

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Last modified August 21 18:08 EDT 2018. Contains 313955 sequences. (Running on oeis4.)