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A131522
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Triangular sequence from coefficients of polynomials of a type gives by a geometric average of two consecutive dihedral group elliptical invariants: p(x,n)=(x^n - 1)*(x^(n + 1) - 1): associate with B_n as odd SO(2*n+1) group.
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0
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1, -1, -1, 1, 1, 0, -1, -1, 0, 1, 1, 0, 0, -1, -1, 0, 0, 1, 1, 0, 0, 0, -1, -1, 0, 0, 0, 1, 1, 0, 0, 0, 0, -1, -1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| Derivation: dihedral invariant given by Singerman
jD_n(x,n)=(x^n - 1)^2/(-4*x^n)
Geometric average as B_n Cartan elliptical invariant estimate:
jB_n(x,n)=Sqrt[jD_n(x,n)*jD_n(x,n+1)]=p(x,n)/(4*x^((2*n+1)/2))
Expansion sequence:
p[x_, n_] = ExpandAll[(x^n - 1)*(x^(n + 1) - 1)];
(* toral inverse the same as the polynomial*)
Table[Table[ SeriesCoefficient[Series[1/p[x, m], {x, 0, 30}], n], {n, 0, 30}], {m, 1, 5}]
m = 1; A004526, m = 2; A008615, m = 3; A008679, m = 4; unknown, m = 5; A033182
They might be called "pyramidal numbers", except that term has a different meaning; they tie four known sequence and maybe more together.
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REFERENCES
| Gareth Jones and David Singerman, Bull. London Math Soc. 28, (1996) pages 561-590 ( dihedral group invariant on page 585)
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FORMULA
| a(n,m) = CoefficientList[x^(2*n+1)-x^(n+1)-x^n+1,x]
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EXAMPLE
| {1, -1, -1, 1},
{1, 0, -1, -1, 0, 1},
{1, 0,0, -1, -1, 0, 0, 1},
{1, 0, 0, 0, -1, -1, 0, 0, 0, 1},
{1, 0, 0, 0, 0, -1, -1, 0, 0, 0, 0, 1},
{1, 0, 0, 0, 0, 0, -1, -1,0, 0, 0, 0, 0, 1},
{1, 0, 0, 0, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0,0, 1},
{1, 0, 0, 0, 0, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0, 0, 0, 1},
{1, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1},
{1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1}
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MATHEMATICA
| p[n_] = ExpandAll[(x^n - 1)*(x^(n + 1) - 1)]; a = Table[CoefficientList[p[n], x], {n, 1, 10}]; Flatten[a]
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CROSSREFS
| Cf. A004526, A008615, A008679, A033182.
Sequence in context: A071036 A118110 A071038 * A144473 A011750 A010055
Adjacent sequences: A131519 A131520 A131521 * A131523 A131524 A131525
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KEYWORD
| tabl,uned,sign
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AUTHOR
| Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Aug 24 2007
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