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%I #4 Jan 06 2017 10:53:49
%S 1,2,5,13,34,1,89,7,1,233,34,7,1,610,141,35,7,1,1597,534,152,36,7,1,
%T 4181,1905,611,163,37,7,1,10946,6512,2338,689,174,38,7,1,28657,21557,
%U 8641,2787,768,185,39,7,1,75025,69593,31104,10921,3252,848,196,40,7,1
%N Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having k horizontal steps in the valleys (n>=2, k>=0).
%C Number of entries in rows 2,3,4,5 is 1; number of entries in row n (n>=5) is n-4.
%C Sum of entries in row n = A082582(n).
%C T(n,0) = A001519(n-1) = F(2n-3), where F(n) are the Fibonacci numbers A000045.
%C Sum(k*T(n,k), k>=0) = A278135(n).
%H A. Blecher, C. Brennan, and A. Knopfmacher, <a href="http://dx.doi.org/10.1080/0035919X.2015.1059905">Peaks in bargraphs</a>, Trans. Royal Soc. South Africa, 71, No. 1, 2016, 97-103.
%F G.f.: G(t,z), where t marks number of horizontal steps in the valleys and z marks semiperimeter, satisfies aG^2 - bG + c = 0, where a = tz(1-z)^2, b = 1 - 3z - tz + z^2 + 3t*z^2 -tz^4, c = z^2*(1-z)(1-tz).
%e Row 6 is 34,1 because among the 35 (=A082582(6)) bargraphs of semiperimeter 6 only one has a valley; it corresponds to the composition [2,1,2] and its width is 1.
%e Triangle starts:
%e 1;
%e 2;
%e 5;
%e 13;
%e 34, 1;
%e 89, 7, 1
%p a := t*z*(1-z)^2: b := 1-3*z-t*z+z^2+3*t*z^2-t*z^4: c := z^2*(1-z)*(1-t*z): G := RootOf(a*G^2-b*G+c = 0, G): Gser := simplify(series(G, z = 0, 20)): for n from 2 to 16 do P[n] := sort(coeff(Gser, z, n)) end do: 1; 2; 5; 13; for n from 6 to 16 do seq(coeff(P[n], t, j), j = 0 .. n-5) end do; # yields sequence in triangular form
%Y Cf. A001519, A082582, A273719, A273720, A278135
%K nonn,tabf
%O 2,2
%A _Emeric Deutsch_, Jan 06 2017
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