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A140284
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A weighted binomial triangle with smooth weighting function: f(n,d) = Floor[1 + d*Sech[d/2 - n]].
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0
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1, 1, 1, 1, 6, 1, 1, 9, 9, 1, 1, 12, 30, 12, 1, 1, 15, 50, 50, 15, 1, 1, 12, 60, 140, 60, 12, 1, 1, 14, 63, 245, 245, 63, 14, 1, 1, 8, 84, 336, 630, 336, 84, 8, 1, 1, 9, 72, 336, 1008, 1008, 336, 72, 9, 1, 1, 10, 45, 360, 1470, 2772, 1470, 360, 45, 10, 1
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OFFSET
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1,5
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COMMENTS
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Row sums are:
{1, 2, 4, 13, 46, 118, 275, 633, 1481, 2844, 6535}.;
Here the interior coefficients are larger than the
pascal triangle: most generalized Pascal triangles yield smaller interior
coefficients.
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LINKS
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FORMULA
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a(n,d)=If[n == 0 || n == d, 1, f[n, d]* Binomial[d, n]]; f(n,d) = Floor[1 + d*Sech[d/2 - n]].
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EXAMPLE
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{{1},
{1, 1},
{1, 6, 1},
{1, 9, 9, 1},
{1, 12, 30, 12, 1},
{1, 15, 50, 50, 15,1},
{1, 12, 60, 140, 60, 12, 1},
{1, 14, 63, 245, 245, 63, 14, 1},
{1, 8, 84, 336, 630, 336, 84, 8, 1},
{1, 9, 72, 336, 1008, 1008, 336, 72, 9,1},
{1, 10, 45, 360, 1470, 2772, 1470, 360, 45, 10, 1}}
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MATHEMATICA
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f[n_, d_] = Floor[1 + d*Sech[d/2 - n]]; a = Table[Table[If[n == 0 || n == d, 1, f[n, d]* Binomial[ d, n]], {n, 0, d}], {d, 0, 10}]; Flatten[a] Table[Apply[Plus, Table[If[n == 1 || n == d, 1, f[n, d]* Binomial[d, n]], {n, 0, d}]], {d, 0, 10}];
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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