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A278132
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Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having height k (n>=2, 1<=k<=n-1).
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1
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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
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2,5
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COMMENTS
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Number of entries in row n is n-1.
Sum of entries in row n = A082582(n).
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LINKS
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FORMULA
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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.
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EXAMPLE
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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.
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MAPLE
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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
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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