%I
%S 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,
%T 64,219,258,172,66,13,1,1,113,513,721,577,295,91,15,1,1,199,1184,1975,
%U 1817,1125,466,120,17,1,1,350,2702,5326,5534,3932,1994,693,153,19,1
%N Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having height k (n>=2, 1<=k<=n1).
%C Number of entries in row n is n1.
%C Sum of entries in row n = A082582(n).
%C Sum(k*T(n,k),k>=0) = A278133(n)
%D A. Blecher, C. Brennan, A. Knopfmacher, The height and width of bargraphs, Discrete Appl. Math., 180, 2015, 3644.
%F 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.
%e 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.
%p 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
%Y Cf. A278133
%K nonn,tabl
%O 2,5
%A _Emeric Deutsch_, Dec 31 2016
