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Triangle generated by T(n,k) = Fibonacci(n-k+2)^k.
5

%I #39 Mar 16 2023 12:23:09

%S 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,

%T 21,169,512,625,243,64,1,1,34,441,2197,4096,3125,729,128,1,1,55,1156,

%U 9261,28561,32768,15625,2187,256,1,1,89,3025,39304,194481,371293

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

%C 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

%C Row sums = A210342

%C Central coefficients = A067966.

%C This triangle is related to the infinite Vandermonde matrix

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

%C 1, 1, 1, 1, 1, 1, 1

%C 1, 2, 4, 8, 16, 32, 64

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

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

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

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

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

%C The generating series of the columns can be expressed in terms of Fibonomial coefficients (A010048) (see Riordan's paper).

%H Vincenzo Librandi, <a href="/A210341/b210341.txt">Rows n = 0..90, flattened</a>

%H J. Riordan, <a href="http://dx.doi.org/10.1215/S0012-7094-62-02902-2">Generating functions for powers of Fibonacci numbers</a>, Duke. Math. J. 29 (1962) 5-12.

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

%e Triangle begins:

%e 1

%e 1, 1

%e 1, 2, 1

%e 1, 3, 4, 1

%e 1, 5, 9, 8, 1

%e 1, 8, 25, 27, 16, 1

%e 1, 13, 64, 125, 81, 32, 1

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

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

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

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

%o (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

%Y Cf. A103323, A067966, A210342, A210343.

%K nonn,tabl,easy

%O 0,5

%A _Emanuele Munarini_, Mar 20 2012