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A140818
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Coefficients of Hodge diamond binomial 'X' matrices as polynomials: matrix example; M={{1,0,1}. {0,2,0], {1,0,1}: M(d, x, y)= Sum[Sum[If[n == m, Binomial[d - 1, m - 1], If[n == d - m + 1, Binomial[d - 1, n - 1], 0]]*x^(n - 1)*y^(m - 1), {n, 1, d}], {m, 1, d}] .
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0
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1, 2, 2, 2, 2, 2, 2, 6, 6, 2, 2, 8, 6, 8, 2, 2, 10, 20, 20, 10, 2, 2, 12, 30, 20, 30, 12, 2, 2, 14, 42, 70, 70, 42, 14, 2, 2, 16, 56, 112, 70, 112, 56, 16, 2, 2, 18, 72, 168, 252, 252, 168, 72, 18, 2
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OFFSET
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1,2
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COMMENTS
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LINKS
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FORMULA
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M(d, x, y)= Sum[Sum[If[n == m, Binomial[d - 1, m - 1], If[n == d - m + 1, Binomial[d - 1, n - 1], 0]]*x^(n - 1)*y^(m - 1), {n, 1, d}], {m, 1, d}] ; a(n,m)=Coefficients(M(n,x,1)).
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EXAMPLE
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{1},
{2, 2},
{2, 2, 2},
{2, 6, 6, 2},
{2, 8, 6, 8, 2},
{2, 10, 20, 20, 10, 2},
{2, 12, 30, 20, 30, 12, 2},
{2, 14, 42, 70, 70, 42, 14, 2},
{2, 16, 56, 112, 70, 112, 56, 16, 2},
{2, 18, 72, 168, 252, 252, 168, 72, 18, 2}.
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MATHEMATICA
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Clear[M, y, x] M[d_, x_, y_] := Sum[Sum[If[n == m, Binomial[d - 1, m - 1], If[n == d - m + 1, Binomial[ d - 1, n - 1], 0]]*x^(n - 1)*y^(m - 1), {n, 1, d}], {m, 1, d}] Table[CoefficientList[M[d, x, 1], x], {d, 1, 10}] Flatten[%] Table[Apply[Plus, CoefficientList[M[d, x, 1], x]], {d, 1, 10}]
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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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