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Search: id:A140821
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| A140821 |
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Coefficients of Hodge diamond GCD binomial product 'X' matrices as polynomials: matrix example; M={{2,0,2}. {0,2,0], {2,0,2}: M(d, x, y)= Sum[Sum[If[n == m, GCD[d - 1, m - 1], If[n == d - m + 1, GCD[d - 1, n - 1], 0]]*x^(n - 1)*y^(m - 1), {n, 1, d}], {m, 1, d}] . |
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| 2, 2, 4, 2, 4, 6, 6, 6, 6, 8, 8, 12, 8, 8, 10, 10, 20, 20, 10, 10, 12, 12, 60, 60, 60, 12, 12, 14, 14, 42, 70, 70, 42, 14, 14, 16, 16, 112, 112, 280, 112, 112, 16, 16, 18, 18, 72, 504, 252, 252, 504, 72, 18, 18
(list; table; graph; listen)
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OFFSET
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1,1
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COMMENT
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Row sums are:
{0, 4, 10, 24, 44, 80, 228, 280, 792, 1728}.
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FORMULA
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M(d, x, y)=Sum[Sum[If[n == m, Binomial[d - 1, m - 1]* GCD[d - 1, m - 1], If[n == d - m + 1, Binomial[d - 1, n - 1] *GCD[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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{},
{2, 2},
4, 2, 4},
{6, 6, 6, 6},
{8, 8, 12, 8, 8},
{10, 10, 20, 20, 10, 10},
{12, 12, 60, 60, 60, 12, 12},
{14, 14, 42, 70, 70, 42, 14, 14},
16, 16, 112, 112, 280, 112, 112, 16, 16},
{18, 18, 72, 504, 252, 252, 504, 72, 18, 18}
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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]* GCD[d - 1, m - 1], If[n == d - m + 1, Binomial[d - 1, n - 1] *GCD[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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Cf. A140685.
Sequence in context: A109526 A059214 A091820 this_sequence A063789 A106264 A152423
Adjacent sequences: A140818 A140819 A140820 this_sequence A140822 A140823 A140824
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KEYWORD
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nonn,uned,tabl
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AUTHOR
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Roger L. Bagula and Mats Granvik (rlbagulatftn(AT)yahoo.com), Jul 16 2008
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