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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: 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 May 24 00:57 EDT 2018. Contains 304500 sequences. (Running on oeis4.)