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A273350
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Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having k up steps (n >= 2, k >= 1).
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3
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1, 1, 1, 1, 3, 1, 1, 6, 5, 1, 1, 10, 16, 7, 1, 1, 15, 40, 31, 9, 1, 1, 21, 85, 105, 51, 11, 1, 1, 28, 161, 295, 219, 76, 13, 1, 1, 36, 280, 721, 771, 396, 106, 15, 1, 1, 45, 456, 1582, 2331, 1681, 650, 141, 17, 1, 1, 55, 705, 3186, 6244, 6083, 3235, 995, 181, 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).
Connection with A145904 should be explored.
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LINKS
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FORMULA
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G.f.: G=G(t,z) satisfies zG^2-(1-z-yz-yz^2)G+yz^2=0 (z marks semiperimeter, y marks up-steps).
T(n,m) = (1/(n-m))*Sum_{i=0..m} C(n-m,i-1)*C(n-m,i)*C(n-i-1,m-i). - Vladimir Kruchinin, Jan 15 2018
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EXAMPLE
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Row 4 is 1,3,1 because the 5 (=A082582(4)) bargraphs of semiperimeter 4 correspond to the compositions [1,1,1], [1,2], [2,1], [2,2], [3] which, clearly, have 1,2,2,2,3 up steps.
Triangle starts
1;
1, 1;
1, 3, 1;
1, 6, 5, 1;
1, 10, 16, 7, 1;
1, 15, 40, 31, 9, 1.
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MAPLE
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eq := z*G^2-(1-z-y*z-y*z^2)*G+y*z^2 = 0: G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 18)): for n from 2 to 16 do P[n] := sort(expand(coeff(Gser, z, n))) end do: for n from 2 to 16 do seq(coeff(P[n], y, j), j = 1 .. n-1) end do; # yields sequence in triangular form
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MATHEMATICA
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G = (-1 + z + y z + y z^2 + Sqrt[-4y z^3 + (-1 + z + y z + y z^2)^2])/(2z);
Rest @ CoefficientList[#, y]& /@ Drop[-CoefficientList[G + O[z]^13, z], 2] // Flatten (* Jean-François Alcover, Oct 24 2019 *)
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PROG
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(Maxima)
T(n, m):=if n<=m then 0 else 1/(n-m)*sum(binomial(n-m, i-1)*binomial(n-m, i)*binomial(n-i-1, m-i), i, 0, m); /* Vladimir Kruchinin, Jan 15 2018 */
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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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