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

%I A210341 #39 Mar 16 2023 12:23:09
%S A210341 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 A210341 21,169,512,625,243,64,1,1,34,441,2197,4096,3125,729,128,1,1,55,1156,
%U A210341 9261,28561,32768,15625,2187,256,1,1,89,3025,39304,194481,371293
%N A210341 Triangle generated by T(n,k) = Fibonacci(n-k+2)^k.
%C A210341 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 A210341 Row sums = A210342
%C A210341 Central coefficients = A067966.
%C A210341 This triangle is related to the infinite Vandermonde matrix
%C A210341 V = [F(i+2)^j]_(i,j>=0) generated by Fibonacci numbers:
%C A210341   1, 1,  1,   1,    1,      1,       1
%C A210341   1, 2,  4,   8,    16,     32,      64
%C A210341   1, 3,  9,   27,   81,     243,     729
%C A210341   1, 5,  25,  125,  625,    3125,    15625
%C A210341   1, 8,  64,  512,  4096,   32768,   262144
%C A210341   1, 13, 169, 2197, 28561,  371293,  4826809
%C A210341  1, 21, 441, 9261, 194481, 4084101, 85766121
%C A210341 The generating series of the columns can be expressed in terms of Fibonomial coefficients (A010048) (see Riordan's paper).
%H A210341 Vincenzo Librandi, <a href="/A210341/b210341.txt">Rows n = 0..90, flattened</a>
%H A210341 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 A210341 G.f.: Sum_{k>=0} x^k/(1-Fibonacci(k+2)*x*y).
%e A210341 Triangle begins:
%e A210341   1
%e A210341   1,  1
%e A210341   1,  2,   1
%e A210341   1,  3,   4,    1
%e A210341   1,  5,   9,    8,    1
%e A210341   1,  8,  25,   27,   16,    1
%e A210341   1, 13,  64,  125,   81,   32,   1
%e A210341   1, 21, 169,  512,  625,  243,  64,   1
%e A210341   1, 34, 441, 2197, 4096, 3125, 729, 128, 1
%t A210341 Flatten[Table[Fibonacci[n-k+2]^k,{n,0,20},{k,0,n}]]
%o A210341 (Maxima) create_list(fib(n-k+2)^k,n,0,20,k,0,n);
%o A210341 (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 A210341 Cf. A103323, A067966, A210342, A210343.
%K A210341 nonn,tabl,easy
%O A210341 0,5
%A A210341 _Emanuele Munarini_, Mar 20 2012