A210341 Triangle generated by T(n,k) = Fibonacci(n-k+2)^k.

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

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

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).

Links

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

J. Riordan, Generating functions for powers of Fibonacci numbers, Duke. Math. J. 29 (1962) 5-12.

Formula

G.f.: Sum_{k>=0} x^k/(1-Fibonacci(k+2)*x*y).

Example

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

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