

A278132


Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having height k (n>=2, 1<=k<=n1).


1



1, 1, 1, 1, 3, 1, 1, 6, 5, 1, 1, 11, 15, 7, 1, 1, 20, 38, 28, 9, 1, 1, 36, 92, 89, 45, 11, 1, 1, 64, 219, 258, 172, 66, 13, 1, 1, 113, 513, 721, 577, 295, 91, 15, 1, 1, 199, 1184, 1975, 1817, 1125, 466, 120, 17, 1, 1, 350, 2702, 5326, 5534, 3932, 1994, 693, 153, 19, 1
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OFFSET

2,5


COMMENTS

Number of entries in row n is n1.
Sum of entries in row n = A082582(n).
Sum(k*T(n,k),k>=0) = A278133(n)


REFERENCES

A. Blecher, C. Brennan, A. Knopfmacher, The height and width of bargraphs, Discrete Appl. Math., 180, 2015, 3644.


LINKS

Table of n, a(n) for n=2..67.


FORMULA

Formula (explained on the Maple program): eq is the recursion equation given in Sec. 2 of the Blecher et al. reference; ic is the initial condition; the resulting g[j]'s agree with the generating functions given in the table on p. 39 of the Blecher et al. reference; H[j]=g[j]g[j1]; Hser[j] is the series expansion of H[j], yielding the entries in column j of the triangle T.


EXAMPLE

T(4,2)=3; indeed, the bargraphs of semiperimeter 4 correspond to the compositions [3], [1,2], [2,2], [2,1], [1,1,1], three of which have height 2.


MAPLE

x := z: y := z: eq := G(h) = x*(y+G(h))+y*G(h1)+x*(y+G(h))*G(h1): ic := G(1) = x*y/(1x): sol := simplify(rsolve({eq, ic}, G(h))): for j to 17 do g[j] := factor(simplify(rationalize(simplify(subs(h = j, sol))))) end do: H[1] := x*y/(1x): for j from 2 to 17 do H[j] := factor(g[j]g[j1]) end do: for j to 17 do Hser[j] := series(H[j], z = 0, 20) end do: T := proc (n, k) coeff(Hser[k], z, n) end proc: for n from 2 to 15 do seq(T(n, k), k = 1 .. n1) end do; # yields sequence in triangular form


CROSSREFS

Cf. A278133
Sequence in context: A055898 A145904 A273350 * A203950 A273349 A159572
Adjacent sequences: A278129 A278130 A278131 * A278133 A278134 A278135


KEYWORD

nonn,tabl


AUTHOR

Emeric Deutsch, Dec 31 2016


STATUS

approved



