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

0,1

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

Table of n, a(n) for n=0..41.

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

A060187, A008292

Sequence in context: A268499 A298522 A206436 * A179990 A228156 A182524

Adjacent sequences: A146540 A146541 A146542 * A146544 A146545 A146546

Keyword

nonn

Author

Roger L. Bagula, Oct 31 2008

Status

approved