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 A146543 The LerchPhi functional part of A060187 MacMahon numbers is treated/ solved for as a curvature to give a set of polynomial triangle sequence coefficients: p(x,n)=Sum[A060187(n,m)*x^(m-1),{m,0,n}]; q(x,n)=k from Solve[FullSimplify[ExpandAll[p[x, n]/(x - 1)^n]] - (1 + k/x^2) == 0, k]. 0
 2, 0, 8, 2, 20, 26, 0, 80, 224, 80, 2, 232, 1692, 1672, 242, 0, 728, 10528, 23568, 10528, 728, 2, 2172, 60678, 259688, 259758, 60636, 2186, 0, 6560, 331584, 2485344, 4674944, 2485344, 331584, 6560, 2, 19664, 1756376, 21707888, 69413420, 69413168 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS The concept here is that the increase in curvature causes transformation of Pascal's triangle into the Eulerian numbers and the MacMahon numbers, while leaving the numerical Modulo 2 Sierpinski Self -Similarity intact. The resulting polynomials have a finite Blaschke elliptical structure. The row sums are: {0, 2, 8, 48, 384, 3840, 46080, 645120, 10321920, 185794560, 3715891200}. REFERENCES Kenneth Hoffman, Banach Spaces of Analytic Functions, Dover, New York, 1962, page 66, page 132. Lennart Carleson and Theodore W. Gamelin, Complex Dynamics, Springer,New York,1993,pp 103 ( Herman's Rings as Finite Blaschke sets) LINKS FORMULA p(x,n)=Sum[A060187(n,m)*x^(m-1),{m,0,n}]; q(x,n)=k from Solve[FullSimplify[ExpandAll[p[x, n]/(x - 1)^n]] - (1 + k/x^2) == 0, k]; t(n,m)=Coefficients(((x - 1)^n/x^2)*q(n,x)). EXAMPLE {}, {2}, {0, 8}, {2, 20, 26}, {0, 80, 224, 80}, {2, 232, 1692, 1672, 242}, {0, 728, 10528, 23568, 10528, 728}, {2, 2172, 60678, 259688, 259758, 60636, 2186}, {0, 6560, 331584, 2485344, 4674944, 2485344, 331584, 6560}, {2, 19664, 1756376, 21707888, 69413420, 69413168, 21708056, 1756304, 19682}, {0, 59048, 9116096, 178301024, 906923072, 1527092720, 906923072, 178301024,9116096, 59048} MATHEMATICA Clear[q, p, x, n, a]; p[x_, n_] = p[x_, n_] = (1 - x)^(n + 1)*PolyLog[ -n, x]/x; q[x_, n_] := ((x - 1)^n/x^2)*k /. Solve[FullSimplify[ExpandAll[p[x, n]/(x - 1)^n]] - (1 + k/x^2) == 0, k]; Table[FullSimplify[Expand[q[x, n]]], {n, 0, 10}]; Table[Flatten[CoefficientList[FullSimplify[Expand[q[x, n]]], x]], {n, 0, 10}]; Flatten[%] CROSSREFS Sequence in context: A159810 A199268 A206436 * A179990 A182524 A212832 Adjacent sequences:  A146540 A146541 A146542 * A146544 A146545 A146546 KEYWORD nonn AUTHOR Roger L. Bagula, Oct 31 2008 STATUS approved

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Last modified May 21 19:04 EDT 2013. Contains 225504 sequences.