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A055858
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Coefficient triangle for certain polynomials.
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11
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1, 1, 2, 4, 9, 6, 27, 64, 48, 36, 256, 625, 500, 400, 320, 3125, 7776, 6480, 5400, 4500, 3750, 46656, 117649, 100842, 86436, 74088, 63504, 54432, 823543, 2097152, 1835008, 1605632, 1404928, 1229312, 1075648, 941192, 16777216, 43046721
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OFFSET
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0,3
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COMMENTS
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The coefficients of the partner polynomials are found in triangle A055864.
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LINKS
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FORMULA
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a(n, m)=0 if n < m; a(0, 0)=1, a(n, 0) = n^n, n >= 1, a(n, m) = n^(m-1)*(n+1)^(n-m+1), n >= m >= 1;
E.g.f. for column m: A(m, x); A(0, x) = 1/(1+W(-x)); A(1, x) = -1 - (d/dx)W(-x) = -1-W(-x)/((1+W(-x))*x); A(2, x) = A(1, x)-int(A(1, x), x)/x-(1/x+x); recursion: A(m, x) = A(m-1, x)-int(A(m-1, x), x)/x-((m-1)^(m-1))*(x^(m-1))/(m-1)!, m >= 3; W(x) principal branch of Lambert's function.
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EXAMPLE
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{1}; {1,2}; {4,9,6}; {27,64,48,36}; ...
Fourth row polynomial (n=3): p(3,x) = 27 + 64*x + 48*x^2 + 36*x^3.
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MATHEMATICA
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a[n_, m_] /; n < m = 0; a[0, 0] = 1; a[n_, 0] := n^n; a[n_, m_] := n^(m-1)*(n+1)^(n-m+1); Table[a[n, m], {n, 0, 8}, {m, 0, n}] // Flatten (* Jean-François Alcover, Jun 20 2013 *)
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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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