|
|
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
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
|
|
|
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
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|