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].

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

Offset: 0

Author

Roger L. Bagula, Oct 31 2008

Keywords

• nonn

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}.

Examples

			{}, {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}
		

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)

Crossrefs

A060187, A008292

Programs

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[%]

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)).