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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-Fran├žois 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: A322264 A009998 A113993 * A329332 A092056 A103574

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 September 24 21:25 EDT 2020. Contains 337322 sequences. (Running on oeis4.)