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Search: id:A139548
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| A139548 |
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A triangle of coefficients of a product of Pascal binomials in three variables {x,y,z} reduced to one variable: 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(z,x,n);. |
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+0
1
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| 1, 2, 4, 2, 4, 16, 24, 16, 4, 8, 48, 120, 160, 120, 48, 8, 16, 128, 448, 896, 1120, 896, 448, 128, 16, 32, 320, 1440, 3840, 6720, 8064, 6720, 3840, 1440, 320, 32, 64, 768, 4224, 14080, 31680, 50688, 59136, 50688, 31680, 14080, 4224, 768, 64, 128, 1792, 11648
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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Row sums are:A001018
{1, 8, 64, 512, 4096, 32768, 262144, 2097152, 16777216, 134217728, 1073741824}
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FORMULA
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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(z,x,n); Out_n,m=Coefficients(p(x,1,1,n)).
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EXAMPLE
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{1},
{2, 4, 2},
{4, 16, 24, 16, 4},
{8, 48, 120, 160, 120, 48, 8},
{16, 128, 448, 896, 1120, 896, 448, 128, 16},
{32, 320, 1440, 3840, 6720, 8064, 6720, 3840, 1440, 320, 32},
{64, 768, 4224, 14080, 31680, 50688, 59136, 50688, 31680, 14080, 4224, 768, 64},
{128, 1792, 11648, 46592, 128128, 256256, 384384, 439296, 384384, 256256, 128128, 46592, 11648, 1792, 128},
{256, 4096, 30720, 143360, 465920, 1118208, 2050048, 2928640, 3294720, 2928640, 2050048, 1118208, 465920, 143360, 30720, 4096, 256}, {512, 9216, 78336, 417792, 1566720, 4386816, 9504768, 16293888, 22404096, 24893440, 22404096, 16293888, 9504768, 4386816, 1566720, 417792, 78336, 9216, 512},
{1024, 20480, 194560, 1167360, 4961280, 15876096, 39690240, 79380480, 128993280, 171991040, 189190144, 171991040, 128993280, 79380480, 39690240, 15876096, 4961280, 1167360, 194560, 20480, 1024}
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MATHEMATICA
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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
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Cf. A001018.
Sequence in context: A151706 A055372 A136620 this_sequence A108445 A019294 A057037
Adjacent sequences: A139545 A139546 A139547 this_sequence A139549 A139550 A139551
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KEYWORD
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nonn,uned
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AUTHOR
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Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Jun 10 2008
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