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A139585
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A triangle of coefficients from truncated Pascal binomials in {x,y},y,z}and {z,x} in which the x^n,y^n and z^n terms are deleted: f(x,y,n)=(x+y)^n-(x^n+y^n); p(x,y,z,n)=f(x,y,n)+f(y,z,n)+f(z,x,n).
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0
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-3, 0, 2, 4, 6, 6, 6, 14, 8, 12, 8, 30, 10, 20, 20, 10, 62, 12, 30, 40, 30, 12, 126, 14, 42, 70, 70, 42, 14, 254, 16, 56, 112, 140, 112, 56, 16, 510, 18, 72, 168, 252, 252, 168, 72, 18, 1022, 20, 90, 240, 420, 504, 420, 240, 90, 20
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| Row sums are:
{-3, 0, 6, 18, 42, 90, 186, 378, 762, 1530, 3066};
These polynomials are inspired by the looping hyperfolium projections of the type:
f[x,i,n)=Binomial[n,i]*x^i/(1+x^n);
in n-1 coordinates.
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FORMULA
| f(x,y,n)=(x+y)^n-(x^n+y^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
| {-3},
{0},
{2, 4},
{6, 6, 6},
{14, 8, 12, 8},
{30, 10, 20, 20, 10},
{62, 12, 30, 40, 30, 12},
{126, 14, 42, 70, 70, 42, 14},
{254, 16, 56, 112, 140, 112, 56, 16},
{510, 18, 72, 168, 252, 252, 168, 72, 18},
{1022, 20, 90, 240, 420, 504, 420, 240, 90, 20}
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MATHEMATICA
| f[x_, y_, n_] := (x + y)^n - x^n - y^n 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
| Sequence in context: A145856 A092154 A177344 * A089598 A117139 A159959
Adjacent sequences: A139582 A139583 A139584 * A139586 A139587 A139588
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KEYWORD
| uned,sign
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AUTHOR
| Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Jun 11 2008
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