%I #15 Sep 14 2024 13:23:32
%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<=n-1).
%C Number of entries in row n is n-1.
%C Sum of entries in row n = A082582(n).
%C Sum_{k>=0} k*T(n,k) = A278133(n).
%H A. Blecher, C. Brennan, A. Knopfmacher, and H. Prodinger, <a href="https://doi.org/10.1016/j.dam.2014.08.026">The height and width of bargraphs</a>, Discrete Applied Math. 180, (2015), 36-44.
%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[j-1]; 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.
%e Triangle begins:
%e 1;
%e 1, 1;
%e 1, 3, 1;
%e 1, 6, 5, 1;
%e 1, 11, 15, 7, 1;
%e ...
%p x := z: y := z: eq := G(h) = x*(y+G(h))+y*G(h-1)+x*(y+G(h))*G(h-1): ic := G(1) = x*y/(1-x): 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/(1-x): for j from 2 to 17 do H[j] := factor(g[j]-g[j-1]) 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 .. n-1) end do; # yields sequence in triangular form
%t x = y = z;
%t eq = G[h] == x*(y + G[h]) + y*G[h - 1] + x*(y + G[h])*G[h - 1];
%t ic = G[1] == x*y/(1 - x);
%t sol = RSolve[{eq, ic}, G[h], h];
%t For[j = 1, j <= 17, j++, g[j] = G[h] /. sol /. h -> j];
%t H[1] = x*y/(1 - x);
%t For[j = 2, j <= 17, j++, H[j] = g[j] - g[j - 1]];
%t For[j = 1, j <= 17, j++, Hser[j] = Series[H[j][[1]], {z, 0, 20}]];
%t T[n_, k_] := Coefficient[Hser[k], z, n];
%t Table[T[n, k], {n, 2, 15}, {k, 1, n - 1}] // Flatten (* _Jean-François Alcover_, Sep 14 2024, after Maple program *)
%Y Cf. A082582, A278133.
%K nonn,tabl
%O 2,5
%A _Emeric Deutsch_, Dec 31 2016