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%I #95 Aug 12 2022 12:37:47
%S 0,1,0,1,1,0,1,1,1,0,1,2,1,1,0,1,3,2,1,1,0,1,5,4,2,1,1,0,1,8,7,4,2,1,
%T 1,0,1,13,13,8,4,2,1,1,0,1,21,24,15,8,4,2,1,1,0,1,34,44,29,16,8,4,2,1,
%U 1,0,1,55,81,56,31,16,8,4,2,1,1,0,1,89,149,108,61,32,16,8,4,2,1,1,0
%N Array F(k, n) read by descending antidiagonals: k-generalized Fibonacci numbers in row k >= 1, starting (0, 1, 1, ...), for column n >= 0.
%C For all k >= 1, the k-generalized Fibonacci number F(k,n) satisfies the recurrence obtained by adding more terms to the recurrence of the Fibonacci numbers.
%C The number of tilings of an 1 X n rectangle with tiles of size 1 X 1, 1 X 2, ..., 1 X k is F(k,n).
%C T(k,n) is the number of 0-balanced ordered trees with n edges and height k (height is the number of edges from root to a leaf). - _Emeric Deutsch_, Jan 19 2007
%C Brlek et al. (2006) call this table "number of psp-polyominoes with flat bottom". - _N. J. A. Sloane_, Oct 30 2018
%H Alois P. Heinz, <a href="/A092921/b092921.txt">Antidiagonals n = 0..140, flattened</a>
%H Srecko Brlek, Andrea Frosini, Simone Rinaldi, and Laurent Vuillon, <a href="https://doi.org/10.37236/1041">Tilings by translation: enumeration by a rational language approach</a>, The Electronic Journal of Combinatorics, vol. 13, (2006). Table 1 is essentially this array. - _N. J. A. Sloane_, Jul 20 2014
%H E. S. Egge, <a href="https://arxiv.org/abs/math/0109219">Restricted permutations related to Fibonacci numbers and k-generalized Fibonacci numbers</a>, arXiv:math/0109219 [math.CO], 2001.
%H E. S. Egge, <a href="https://arxiv.org/abs/math/0307050">Restricted 3412-Avoiding Involutions</a>, arXiv:math/0307050 [math.CO], 2003.
%H E. S. Egge and T. Mansour, <a href="https://arxiv.org/abs/math/0203226">Restricted permutations, Fibonacci numbers and k-generalized Fibonacci numbers</a>, arXiv:math/0203226 [math.CO], 2002.
%H E. S. Egge and T. Mansour, <a href="https://arxiv.org/abs/math/0209255">231-avoiding involutions and Fibonacci numbers</a>, arXiv:math/0209255 [math.CO], 2002.
%H Nathaniel D. Emerson, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL9/Emerson/emerson6.html">A Family of Meta-Fibonacci Sequences Defined by Variable-Order Recursions</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.8.
%H Abraham Flaxman, Aram W. Harrow, and Gregory B. Sorkin, <a href="https://doi.org/10.37236/1761">Strings with Maximally Many Distinct Subsequences and Substrings</a>, Electronic J. Combinatorics 11 (1) (2004), Paper R8.
%H I. Flores, <a href="http://www.fq.math.ca/Scanned/5-3/flores.pdf">k-Generalized Fibonacci numbers</a>, Fib. Quart., 5 (1967), 258-266.
%H H. Gabai, <a href="http://www.fq.math.ca/Scanned/8-1/gabai.pdf">Generalized Fibonacci k-sequences</a>, Fib. Quart., 8 (1970), 31-38.
%H R. Kemp, <a href="http://dx.doi.org/10.1002/rsa.3240050111">Balanced ordered trees</a>, Random Structures and Alg., 5 (1994), pp. 99-121.
%H E. P. Miles jr., <a href="http://www.jstor.org/stable/2308649">Generalized Fibonacci numbers and associated matrices</a>, The Amer. Math. Monthly, 67 (1960) 745-752.
%H M. D. Miller, <a href="http://www.jstor.org/stable/2316316">On generalized Fibonacci numbers</a>, The Amer. Math. Monthly, 78 (1971) 1108-1109.
%H Harold R. Parks and Dean C. Wills, <a href="https://arxiv.org/abs/2208.01224">Sum of k-bonacci Numbers</a>, arXiv:2208.01224 [math.CO], 2022. See p. 5.
%F F(k,n) = F(k,n-1) + F(k,n-2) + ... + F(k,n-k); F(k,1) = 1 and F(k,n) = 0 for n <= 0.
%F G.f.: x/(1-Sum_{i=1..k} x^i).
%F F(k,n) = 2^(n-2) for 1 < n <= k+1. - _M. F. Hasler_, Apr 20 2018
%F F(k,n) = Sum_{j=0..floor(n/(k+1))} (-1)^j*((n - j*k) + j + delta(n,0))/(2*(n - j*k) + delta(n,0))*binomial(n - j*k, j)*2^(n-j*(k+1)), where delta denotes the Kronecker delta (see Corollary 3.2 in Parks and Wills). - _Stefano Spezia_, Aug 06 2022
%e From _Peter Luschny_, Apr 03 2021: (Start)
%e Array begins:
%e n = 0 1 2 3 4 5 6 7 8 9 10
%e -------------------------------------------------------------
%e [k=1, mononacci ] 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e [k=2, Fibonacci ] 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...
%e [k=3, tribonacci] 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, ...
%e [k=4, tetranacci] 0, 1, 1, 2, 4, 8, 15, 29, 56, 108, 208, ...
%e [k=5, pentanacci] 0, 1, 1, 2, 4, 8, 16, 31, 61, 120, 236, ...
%e [k=6] 0, 1, 1, 2, 4, 8, 16, 32, 63, 125, 248, ...
%e [k=7] 0, 1, 1, 2, 4, 8, 16, 32, 64, 127, 253, ...
%e [k=8] 0, 1, 1, 2, 4, 8, 16, 32, 64, 128, 255, ...
%e [k=9] 0, 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, ...
%e Note that the first parameter in F(k, n) refers to rows, and the second parameter refers to columns. This is always the case. Only the usual naming convention for the indices is not adhered to because it is common to call the row sequences k-bonacci numbers. (End)
%e .
%e From _Peter Luschny_, Aug 12 2015: (Start)
%e As a triangle counting compositions of n with largest part k:
%e n\k]| [0][1] [2] [3] [4][5][6][7][8][9]
%e [0] | [0]
%e [1] | [0, 1]
%e [2] | [0, 1, 1]
%e [3] | [0, 1, 1, 1]
%e [4] | [0, 1, 2, 1, 1]
%e [5] | [0, 1, 3, 2, 1, 1]
%e [6] | [0, 1, 5, 4, 2, 1, 1]
%e [7] | [0, 1, 8, 7, 4, 2, 1, 1]
%e [8] | [0, 1, 13, 13, 8, 4, 2, 1, 1]
%e [9] | [0, 1, 21, 24, 15, 8, 4, 2, 1, 1]
%e For example for n=7 and k=3 we have the 7 compositions [3, 3, 1], [3, 2, 2], [3, 2, 1, 1], [3, 1, 3], [3, 1, 2, 1], [3, 1, 1, 2], [3, 1, 1, 1, 1].
%e (End)
%p F:= proc(k, n) option remember; `if`(n<2, n,
%p add(F(k, n-j), j=1..min(k,n)))
%p end:
%p seq(seq(F(k, d+1-k), k=1..d+1), d=0..12); # _Alois P. Heinz_, Nov 02 2016
%p # Based on the above function:
%p Arow := (k, len) -> seq(F(k, j), j = 0..len):
%p seq(lprint(Arow(k, 14)), k = 1..10); # _Peter Luschny_, Apr 03 2021
%t F[k_, n_] := F[k, n] = If[n<2, n, Sum[F[k, n-j], {j, 1, Min[k, n]}]];
%t Table[F[k, d+1-k], {d, 0, 12}, {k, 1, d+1}] // Flatten (* _Jean-François Alcover_, Jan 11 2017, translated from Maple *)
%o (PARI) F(k,n)=if(n<2,if(n<1,0,1),sum(i=1,k,F(k,n-i)))
%o (PARI) T(m,n)=!!n*(matrix(m,m,i,j,j==i+1||i==m)^(n+m-2))[1,m] \\ _M. F. Hasler_, Apr 20 2018
%o (PARI) F(k,n) = if(n==0,0, polcoeff(lift(Mod('x, Pol(vector(k+1,i, if(i==1,1,-1))))^(n+k-2)), k-1)); \\ _Kevin Ryde_, Jun 05 2020
%o (Sage)
%o # As a triangle of compositions of n with largest part k.
%o C = lambda n,k: Compositions(n, max_part=k, inner=[k]).cardinality()
%o for n in (0..9): [C(n,k) for k in (0..n)] # _Peter Luschny_, Aug 12 2015
%Y Columns converge to A166444: each column n converges to A166444(n) = 2^(n-2).
%Y Rows 1-8 are (shifted) A057427, A000045, A000073, A000078, A001591, A001592, A066178, A079262.
%Y Essentially a reflected version of A048887.
%Y See A048004 and A126198 for closely related arrays.
%Y Cf. A066099.
%K nonn,tabl
%O 0,12
%A _Ralf Stephan_, Apr 17 2004
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