A278134 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).

1, 2, 5, 13, 34, 1, 89, 7, 1, 233, 34, 7, 1, 610, 141, 35, 7, 1, 1597, 534, 152, 36, 7, 1, 4181, 1905, 611, 163, 37, 7, 1, 10946, 6512, 2338, 689, 174, 38, 7, 1, 28657, 21557, 8641, 2787, 768, 185, 39, 7, 1, 75025, 69593, 31104, 10921, 3252, 848, 196, 40, 7, 1

Offset

2,2

Comments

Number of entries in rows 2,3,4,5 is 1; number of entries in row n (n>=5) is n-4.

Sum of entries in row n = A082582(n).

T(n,0) = A001519(n-1) = F(2n-3), where F(n) are the Fibonacci numbers A000045.

Sum(k*T(n,k), k>=0) = A278135(n).

Links

Table of n, a(n) for n=2..59.

A. Blecher, C. Brennan, and A. Knopfmacher, Peaks in bargraphs, Trans. Royal Soc. South Africa, 71, No. 1, 2016, 97-103.

Formula

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).

Example

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.
Triangle starts:
1;
2;
5;
13;
34, 1;
89, 7, 1

Maple

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

Crossrefs

Cf. A001519, A082582, A273719, A273720, A278135

Sequence in context: A001429 A148288 A320813 * A271940 A273721 A112841

Adjacent sequences: A278131 A278132 A278133 * A278135 A278136 A278137

Keyword

nonn,tabf

Author

Emeric Deutsch, Jan 06 2017

Status

approved