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A146959
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A functionally symmetric Polynomial as a triangle of coefficients: p(x,n)=If[n == 0, 1, (x + 1)^n + 2^(n - 3)*Sum[(3^(m-1) + 2*m+(n-1) )*x^m*(1 + x^(n - 2*m)), {m, 1, n - 1}]].
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0
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1, 1, 1, 1, 6, 1, 1, 17, 17, 1, 1, 52, 46, 52, 1, 1, 189, 130, 130, 189, 1, 1, 838, 431, 340, 431, 838, 1, 1, 4327, 1781, 1027, 1027, 1781, 4327, 1, 1, 24328, 8860, 3896, 2758, 3896, 8860, 24328, 1, 1, 142217, 49060, 18388, 9214, 9214, 18388, 49060, 142217, 1
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OFFSET
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0,5
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COMMENTS
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Row sums are:{1, 2, 8, 36, 152, 640, 2880, 14272, 76928, 437760, 2564096}.
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LINKS
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FORMULA
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p(x,n)=If[n == 0, 1, (x + 1)^n + 2^(n - 3)*Sum[(3^(m-1) + 2*m+(n-1) )*x^m*(1 + x^(n - 2*m)), {m, 1, n - 1}]]; t(n,m)=coefficients(p(x,n)).
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EXAMPLE
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{1}, {1, 1}, {1, 6, 1}, {1, 17, 17, 1}, {1, 52, 46, 52, 1}, {1, 189, 130, 130, 189, 1}, {1, 838, 431, 340, 431, 838, 1}, {1, 4327, 1781, 1027, 1027, 1781, 4327, 1}, {1, 24328, 8860, 3896, 2758, 3896, 8860, 24328, 1}, {1, 142217, 49060, 18388, 9214, 9214, 18388, 49060, 142217, 1}, {1, 844810, 285229, 99448, 39634, 25852, 39634, 99448, 285229, 844810, 1}
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MATHEMATICA
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Clear[p, x, n]; p[x_, n_] = If[ n == 0, 1, (x + 1)^n + 2^(n - 3)*Sum[(3^(m-1) + 2*m+(n-1) )*x^m*(1 + x^(n - 2*m)), {m, 1, n - 1}]]; Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}]; Flatten[%]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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