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%I #28 Mar 07 2023 11:07:44
%S 1,2,2,1,2,3,2,5,1,2,7,4,2,9,9,1,2,11,16,5,2,13,25,14,1,2,15,36,30,6,
%T 2,17,49,55,20,1,2,19,64,91,50,7,2,21,81,140,105,27,1,2,23,100,204,
%U 196,77,8,2,25,121,285,336,182,35,1,2,27,144,385,540,378,112,9,2,29,169,506
%N Triangle read by rows: T(n,k) is the number of Fibonacci binary words of length n and having k 01 subwords (0 <= k <= floor(n/2)). A Fibonacci binary word is a binary word having no 00 subword.
%C Also number of Fibonacci binary words of length n and having k 10 subwords.
%C Row n has 1+floor(n/2) terms.
%C Row sums are the Fibonacci numbers (A000045).
%C T(n,0)=2 for n >= 1.
%C Sum_{k>=0} k*T(n,k) = A023610(n-2).
%C Triangle, with zeros omitted, given by (2, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Jan 14 2012
%C Riordan array ((1+x)/(1-x), x^2/(1-x)), zeros omitted. - _Philippe Deléham_, Jan 14 2012
%H Michael De Vlieger, <a href="/A129710/b129710.txt">Table of n, a(n) for n = 0..10200</a> (rows 0 <= n <= 200, flattened.)
%H Jean-Luc Baril and José Luis Ramírez, <a href="https://arxiv.org/abs/2302.12741">Descent distribution on Catalan words avoiding ordered pairs of Relations</a>, arXiv:2302.12741 [math.CO], 2023.
%H Thomas Grubb and Frederick Rajasekaran, <a href="https://arxiv.org/abs/2009.00650">Set Partition Patterns and the Dimension Index</a>, arXiv:2009.00650 [math.CO], 2020. Mentions this sequence.
%F T(n,k) = binomial(n-k,k) + binomial(n-k-1,k) for n >= 1 and 0 <= k <= floor(n/2).
%F G.f. = G(t,z) = (1+z)/(1-z-tz^2).
%F Sum_{k=0..n} T(n,k)*x^k = (-1)^n*A078050(n), A057079(n), A040000(n), A000045(n+2), A000079(n), A006138(n), A026597(n), A133407(n), A133467(n), A133469(n), A133479(n), A133558(n), A133577(n), A063092(n) for x = -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 respectively. - _Philippe Deléham_, Jan 14 2012
%F T(n,k) = T(n-1,k) + T(n-2,k-1) with T(0,0)=1, T(1,0)=2, T(1,1)=0 and T(n,k) = 0 if k > n or if k < 0. - _Philippe Deléham_, Jan 14 2012
%e T(5,2)=4 because we have 10101, 01101, 01010 and 01011.
%e Triangle starts:
%e 1;
%e 2;
%e 2, 1;
%e 2, 3;
%e 2, 5, 1;
%e 2, 7, 4;
%e 2, 9, 9, 1;
%e Triangle (2, -1, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, ...) begins:
%e 1;
%e 2, 0;
%e 2, 1, 0;
%e 2, 3, 0, 0;
%e 2, 5, 1, 0, 0;
%e 2, 7, 4, 0, 0, 0;
%e 2, 9, 9, 1, 0, 0, 0;
%p T:=proc(n,k) if n=0 and k=0 then 1 elif k<=floor(n/2) then binomial(n-k,k)+binomial(n-k-1,k) else 0 fi end: for n from 0 to 18 do seq(T(n,k),k=0..floor(n/2)) od; # yields sequence in triangular form
%t MapAt[# - 1 &, #, 1] &@ Table[Binomial[n - k, k] + Binomial[n - k - 1, k], {n, 0, 16}, {k, 0, Floor[n/2]}] // Flatten (* _Michael De Vlieger_, Nov 15 2019 *)
%Y Cf. A000045, A023610.
%Y Cf. A029635, A029653.
%Y Columns: A040000, A005408, A000290, A000330, A002415, A005585, A040977, A050486, A053347, A054333, A054334, A057788.
%K nonn,tabf
%O 0,2
%A _Emeric Deutsch_, May 12 2007
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