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A210341 Triangle generated by T(n,k) = Fibonacci(n-k+2)^k. 5
1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 5, 9, 8, 1, 1, 8, 25, 27, 16, 1, 1, 13, 64, 125, 81, 32, 1, 1, 21, 169, 512, 625, 243, 64, 1, 1, 34, 441, 2197, 4096, 3125, 729, 128, 1, 1, 55, 1156, 9261, 28561, 32768, 15625, 2187, 256, 1, 1, 89, 3025, 39304, 194481, 371293 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Number of tilings of an nXk chessboard using monomers and dimers of a fixed orientation. This is easy to see because the board here consists of k independent strips of length n. - Ralf Stephan, May 22 2014

Triangle begins:

1

1,  1

1,  2,   1

1,  3,   4,    1

1,  5,   9,    8,    1

1,  8,  25,   27,   16,    1

1, 13,  64,  125,   81,   32,   1

1, 21, 169,  512,  625,  243,  64,   1

1, 34, 441, 2197, 4096, 3125, 729, 128, 1

Row sums = A210342

Central coefficients = A067966.

This triangle is related to the infinite Vandermonde matrix

V = [F(i+2)^j]_(i,j>=0) generated by Fibonacci numbers:

1, 1,  1,   1,    1,      1,       1

1, 2,  4,   8,    16,     32,      64

1, 3,  9,   27,   81,     243,     729

1, 5,  25,  125,  625,    3125,    15625

1, 8,  64,  512,  4096,   32768,   262144

1, 13, 169, 2197, 28561,  371293,  4826809

1, 21, 441, 9261, 194481, 4084101, 85766121

The generating series of the columns can be expressed in terms of

Fibonomial coefficients (A010048) (see Riordan's paper).

REFERENCES

John Riordan, Generating functions for powers of Fibonacci numbers,

Duke Math. J. Volume 29, Number 1 (1962), 5-12.

LINKS

Vincenzo Librandi, Rows n = 0..90, flattened

FORMULA

G.f. sum(x^k/(1-Fibonacci(k+2)xy),k=0..infinity).

MATHEMATICA

Flatten[Table[Fibonacci[n-k+2]^k, {n, 0, 20}, {k, 0, n}]]

PROG

(Maxima) create_list(fib(n-k+2)^k, n, 0, 20, k, 0, n);

(MAGMA) [Fibonacci(n-k+2)^k: k in [0..n], n in [0..10]]; /* Alternatively: */ [[Fibonacci(n-k+2)^k: k in [0..n]]: n in [0..8]]; // Bruno Berselli, Mar 28 2012

CROSSREFS

Cf. A103323, A067966, A210342, A210343.

Sequence in context: A171881 A321877 A320251 * A160449 A326322 A089940

Adjacent sequences:  A210338 A210339 A210340 * A210342 A210343 A210344

KEYWORD

nonn,tabl,easy

AUTHOR

Emanuele Munarini, Mar 20 2012

STATUS

approved

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Last modified November 14 02:19 EST 2019. Contains 329108 sequences. (Running on oeis4.)