A119462 - OEIS

%I #25 Jan 05 2025 19:51:38

%S 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 252,168,18,2,90,420,420,90,2,2,110,660,924,330,22,2,132,990,1848,990,

%U 132,2,2,156,1430,3432,2574,572,26,2,182,2002,6006,6006,2002,182,2,2,210,2730

%N 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 Row n contains 1 + floor(n/2) terms.

%C Sum of entries in row n is 2^n (A000079).

%C 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 Muniru A Asiru, <a href="/A119462/b119462.txt">Rows n=0..150, flattened</a>

%H 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 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 T(n,k) = 2*binomial(n,2k) for n >= 1; T(0,0) = 1.

%F T(n,k) = 2*T(n-1,k) - T(n-2,k) + T(n-2,k-1) for n >= 3.

%F G.f.: (1 - z^2 + t*z^2)/(1 - 2*z + z^2 - t*z^2).

%F T(n,0) = 2 for n >= 1.

%F T(n,1) = 2*binomial(n,2) = A002378(n-1).

%F T(n,2) = 2*binomial(n,4) = A034827(n).

%F T(n,k) = 2*A034839(n-1,k) for n >= 1. [Corrected by _Georg Fischer_, May 28 2023]

%F Sum_{k=0..floor(n/2)} k*T(n,k) = A057711(n).

%e T(3,1) = 6 because we have 001, 010, 011, 100, 101 and 110.

%e Triangle starts:

%e   1;

%e   2;

%e   2,  2;

%e   2,  6;

%e   2, 12,  2;

%e   2, 20, 10;

%e   2, 30, 30, 2;

%e   ...

%p 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

%o (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 Cf. A000079, A002378, A034827, A034839, A057711.

%K nonn,tabf

%O 0,2

%A _Emeric Deutsch_, May 21 2006