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Search: id:A139584
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| A139584 |
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A triangle of coefficients of A053122 type binomials {x,y},{y,z} and {z,x}, made using A_n Cartan type matrix characteristic polynomials: an(x,n)=CharacteristicPolynomial(M(A_n,n)); f[x,y,n)=Sum[Coefficients(an[x,n)*x^i*y^(n-i),{i,0,n}]; p(x,y,z,n)=f(x,y,n)+f(y,z,n)+f(z,x,n). |
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1
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| 3, 5, -2, 6, -8, 2, 7, -20, 12, -2, 9, -40, 42, -16, 2, 12, -70, 112, -72, 20, -2, 15, -112, 252, -240, 110, -24, 2, 17, -168, 504, -660, 440, -156, 28, -2, 18, -240, 924, -1584, 1430, -728, 210, -32, 2, 19, -330, 1584, -3432, 4004, -2730, 1120, -272, 36, -2, 21, -440, 2574, -6864, 10010, -8736, 4760, -1632, 342
(list; 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:
{3, 3, 0, -3, -3, 0, 3, 3, 0, -3, -3};
In this sequence an antisymmetric group structure A_n is projected onto commutative polynomials,
so that a 3 dimensional quantum surface system results.
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FORMULA
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an(x,n)=CharacteristicPolynomial(M(A_n,n)); f[x,y,n)=Sum[Coefficients(an[x,n)*x^i*y^(n-i),{i,0,n}]; p(x,y,z,n)=f(x,y,n)+f(y,z,n)+f(z,x,n); Out_n,m=Coefficients(p(x,1,1,n).
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EXAMPLE
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{3},
{5, -2},
{6, -8, 2},
{7, -20,12, -2},
{9, -40, 42, -16, 2},
{12, -70, 112, -72, 20, -2},
{15, -112, 252, -240, 110, -24, 2},
{17, -168, 504, -660, 440, -156, 28, -2},
{18, -240, 924, -1584, 1430, -728, 210, -32, 2},
{19, -330, 1584, -3432, 4004, -2730, 1120, -272,36, -2},
{21, -440, 2574, -6864, 10010, -8736, 4760, -1632, 342, -40, 2}
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MATHEMATICA
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Clear[f, x, n] Clear[M, T, d, a, x, an] T[n_, m_, d_] := If[ n == m, 2, If[n == m - 1 || n == m + 1, -1, 0]]; M[d_] := Table[T[n, m, d], {n, 1, d}, {m, 1, d}]; an[x_, n_] := If[n == 0, 1, CharacteristicPolynomial[M[n], x]] f[x_, y_, n_] := Sum[CoefficientList[an[x, n], x][[i + 1]]*x^i*y^(n - i), {i, 0, Length[CoefficientList[an[x, n], x]] - 1}]; Table[ExpandAll[f[x, y, n] + f[y, z, n] + f[x, z, n]], {n, 0, 10}]; a = Table[CoefficientList[ExpandAll[f[x, y, n] + f[y, z, n] + f[x, z, n]] /. y -> 1 /. z -> 1, x], {n, 0, 10}]; Flatten[a]
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CROSSREFS
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Cf. A053122.
Sequence in context: A073264 A016657 A010782 this_sequence A064790 A113966 A164611
Adjacent sequences: A139581 A139582 A139583 this_sequence A139585 A139586 A139587
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KEYWORD
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uned,sign
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AUTHOR
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Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Jun 11 2008
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