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

Author

Emanuele Munarini, Mar 20 2012

Keywords

• nonn
• tabl
• easy

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).

Examples

			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
		

Links

• Vincenzo Librandi, Rows n = 0..90, flattened
• J. Riordan, Generating functions for powers of Fibonacci numbers, Duke. Math. J. 29 (1962) 5-12.

Crossrefs

Cf. A103323, A067966, A210342, A210343.

Programs

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

Mathematica

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

Maxima

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

Formula

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