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A139524
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A trinomial made into a single varaiabe polynomial coefficient triangle form three Pascal binomials in {x,y},{y,z} and {x,z}: binomials as: f(x,y,n)=Sum[Binomial[n, i]*x^i*y^(n - i), {i, 0, n}]; p(x,y,z,n)=f(x,y,n)+f(y,z,n)+f(x,z,n).
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0
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3, 4, 2, 6, 4, 2, 10, 6, 6, 2, 18, 8, 12, 8, 2, 34, 10, 20, 20, 10, 2, 66, 12, 30, 40, 30, 12, 2, 130, 14, 42, 70, 70, 42, 14, 2, 258, 16, 56, 112, 140, 112, 56, 16, 2, 514, 18, 72, 168, 252, 252, 168, 72, 18, 2, 1026, 20, 90, 240, 420, 504, 420, 240, 90, 20, 2
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| Row sums are A007283.
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REFERENCES
| Advanced Number Theory, Harvey Cohn, Dover Books, 1963, Pages 88-89
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FORMULA
| f(x,y,n)=Sum[Binomial[n, i]*x^i*y^(n - i), {i, 0, n}]; p(x,y,z,n)=f(x,y,n)+f(y,z,n)+f(x,z,n); Out_n,m=Coefficients(p(x,1,1,n)).
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EXAMPLE
| {3},
{4, 2},
{6, 4, 2},
{10, 6, 6, 2},
{18, 8, 12, 8, 2},
{34, 10, 20, 20, 10, 2},
{66, 12, 30, 40, 30, 12, 2},
{130, 14, 42, 70, 70, 42, 14, 2},
{258, 16, 56, 112, 140, 112, 56, 16, 2},
{514, 18, 72, 168, 252, 252, 168, 72, 18, 2},
{1026, 20, 90, 240, 420, 504, 420, 240, 90, 20, 2}
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MATHEMATICA
| Clear[f, x, n] f[x_, y_, n_] = Sum[Binomial[n, i]*x^i*y^(n - i), {i, 0, 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
| Cf. A007283.
Sequence in context: A154570 A145961 A082928 * A108127 A049277 A143052
Adjacent sequences: A139521 A139522 A139523 * A139525 A139526 A139527
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KEYWORD
| nonn,uned
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AUTHOR
| Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Jun 09 2008
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