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A119462 Triangle read by rows: T(n,k) is the number of circular binary words of length n having k occurrences of 01 (0 <= k <= floor(n/2)).

%I A119462 #29 Apr 19 2025 11:06:56
%S A119462 1,2,2,2,2,6,2,12,2,2,20,10,2,30,30,2,2,42,70,14,2,56,140,56,2,2,72,
%T A119462 252,168,18,2,90,420,420,90,2,2,110,660,924,330,22,2,132,990,1848,990,
%U A119462 132,2,2,156,1430,3432,2574,572,26,2,182,2002,6006,6006,2002,182,2,2,210,2730,10010,12870,6006,910,30
%N A119462 Triangle read by rows: T(n,k) is the number of circular binary words of length n having k occurrences of 01 (0 <= k <= floor(n/2)).
%C A119462 Row n contains 1 + floor(n/2) terms.
%C A119462 Sum of entries in row n is 2^n (A000079).
%C A119462 2*binomial(n-1,2k) is also the number of permutations avoiding both 123 and 132 with k valleys, i.e., positions with w[i]>w[i+1]<w[i+2]. - _Lara Pudwell_, Dec 19 2018
%H A119462 Muniru A Asiru, <a href="/A119462/b119462.txt">Rows n=0..150, flattened</a>
%H A119462 M. Bukata, R. Kulwicki, N. Lewandowski, L. Pudwell, J. Roth, and T. Wheeland, <a href="https://arxiv.org/abs/1812.07112">Distributions of Statistics over Pattern-Avoiding Permutations</a>, arXiv preprint arXiv:1812.07112 [math.CO], 2018.
%H A119462 L. Carlitz and R. Scoville, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/15-3/carlitz1.pdf">Zero-one sequences and Fibonacci numbers</a>, Fibonacci Quarterly, 15 (1977), 246-254.
%F A119462 T(n,k) = 2*binomial(n,2k) for n >= 1; T(0,0) = 1.
%F A119462 T(n,k) = 2*T(n-1,k) - T(n-2,k) + T(n-2,k-1) for n >= 3.
%F A119462 G.f.: (1 - z^2 + t*z^2)/(1 - 2*z + z^2 - t*z^2).
%F A119462 T(n,0) = 2 for n >= 1.
%F A119462 T(n,1) = 2*binomial(n,2) = A002378(n-1).
%F A119462 T(n,2) = 2*binomial(n,4) = A034827(n).
%F A119462 T(n,k) = 2*A034839(n-1,k) for n >= 1. [Corrected by _Georg Fischer_, May 28 2023]
%F A119462 Sum_{k=0..floor(n/2)} k*T(n,k) = A057711(n).
%e A119462 T(3,1) = 6 because we have 001, 010, 011, 100, 101 and 110.
%e A119462 Triangle starts:
%e A119462   1;
%e A119462   2;
%e A119462   2,  2;
%e A119462   2,  6;
%e A119462   2, 12,  2;
%e A119462   2, 20, 10;
%e A119462   2, 30, 30, 2;
%e A119462   ...
%p A119462 T:=proc(n,k) if n=0 and k=0 then 1 else 2*binomial(n,2*k) fi end: for n from 0 to 15 do seq(T(n,k),k=0..floor(n/2)) od; # yields sequence in triangular form
%t A119462 T[0,0]:=1; T[n_,k_]:= 2*Binomial[n,2k]; Table[T[n,k],{n,0,15},{k,0,Floor[n/2]}]//Flatten (* _Stefano Spezia_, Apr 19 2025 *)
%o A119462 (GAP) Concatenation([1],Flat(List([1..15],n->List([0..Int(n/2)],k->2*Binomial(n,2*k))))); # _Muniru A Asiru_, Dec 20 2018
%Y A119462 Cf. A000079, A002378, A034827, A034839, A057711.
%K A119462 nonn,tabf
%O A119462 0,2
%A A119462 _Emeric Deutsch_, May 21 2006