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%I #27 Jan 18 2022 10:15:38
%S 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,
%T 64,243,625,512,169,21,1,1,128,729,3125,4096,2197,441,34,1,1,256,2187,
%U 15625,32768,28561,9261,1156,55,1,1,512,6561,78125,262144,371293,194481,39304,3025,89
%N Square array T(n,k) read by antidiagonals: powers of Fibonacci numbers.
%C 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).
%D A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, identity 138.
%H Alois P. Heinz, <a href="/A103323/b103323.txt">Antidiagonals n = 1..100, flattened</a>
%F T(n, k) = A000045(k)^n, n, k > 0.
%F 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}) ] ... ]].
%e Square array T(n,k) begins:
%e 1, 1, 2, 3, 5, 8, ...
%e 1, 1, 4, 9, 25, 64, ...
%e 1, 1, 8, 27, 125, 512, ...
%e 1, 1, 16, 81, 625, 4096, ...
%e 1, 1, 32, 243, 3125, 32768, ...
%e 1, 1, 64, 729, 15625, 262144, ...
%e ...
%p A:= (n, k)-> (<<1|1>, <1|0>>^n)[1, 2]^k:
%p seq(seq(A(n, 1+d-n), n=1..d), d=1..12); # _Alois P. Heinz_, Jun 17 2014
%t 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 *)
%o (PARI) T(n,k)=fibonacci(k)^n
%Y Rows include A000045, A007598, A056570, A056571, A056572, A056573, A056574.
%Y Main diagonal gives A100399.
%Y Cf. A244003.
%Y Cf. A105317, A254719.
%K nonn,tabl,easy
%O 1,6
%A _Ralf Stephan_, Feb 02 2005
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