A146540 The PolyLog functional part of A008292 (the Eulerian numbers) is treated as a curvature to give a set of polynomial triangle sequence coefficients: p(x,n)=Sum[A008292(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, -1, 0, 3, -1, 2, 1, 4, -1, 0, 15, 5, 5, -1, 2, 21, 76, 16, 6, -1, 0, 63, 287, 322, 42, 7, -1, 2, 113, 1212, 2381, 1226, 99, 8, -1, 0, 255, 4265, 15675, 15549, 4349, 219, 9, -1, 2, 493, 14644, 88150, 156316, 88108, 14692, 466, 10, -1, 0, 1023, 47795, 455312

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, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800}.

References

Lennart Carleson and Theodore W. Gamelin, Complex Dynamics, Springer, New York, 1993, pp 103 ( Herman's Rings as Finite Blaschke sets)

Kenneth Hoffman, Banach Spaces of Analytic Functions, Dover, New York, 1962, page 66, page 132.

Links

Table of n, a(n) for n=0..57.

Formula

p(x,n)=Sum[A008292(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

{0}, {2, -1}, {0, 3, -1}, {2, 1, 4, -1}, {0, 15, 5, 5, -1}, {2, 21, 76,16, 6, -1}, {0, 63, 287, 322, 42, 7, -1}, {2, 113, 1212, 2381, 1226, 99, 8, -1}, {0, 255, 4265, 15675, 15549, 4349, 219, 9, -1}, {2, 493, 14644, 88150, 156316, 88108, 14692, 466, 10, -1}, {0, 1023, 47795, 455312, 1310144,1310606, 454982, 47960, 968, 11, -1}

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: A333500 A182720 A347316 * A162922 A246180 A102057

Adjacent sequences: A146537 A146538 A146539 * A146541 A146542 A146543

Keyword

sign

Author

Roger L. Bagula, Oct 31 2008, Nov 01 2008

Status

approved