Real-rooted atlas

A091044 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(-5\,x - 18\,x^{2} - 21\,x^{3} - 8\,x^{4} + 4\,n + 23\,n x + 42\,n x^{2} + 31\,n x^{3} + 8\,n x^{4}\right)\,P_{n-1}(x) + \left(-2\,x - 14\,x^{2} - 30\,x^{3} - 26\,x^{4} - 8\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
141014
2168091187
36448011288202492
42562560916813840738133205
51024128006169614272015777266430442442
64096614403724801164800197577617188805978715895343
71638428672020951048268800189958402534880018160336538084078552824
86553613107201121075253473280155319040280870400308347360187656800484275611046681449
92621445898240578027523234201601137153024260225280038711308803605961600190683522043584805013946564870
101048576262144002896035841859911680768191692821299281920401091801605060732160040911830640191228332003922632451185831775139
11419430411534336014181990401028259840048806559744159105515520363195162624580100812800634880360640453155984800189757344824353036920602476125768116
1216777216503316480681679257655076454400295445790720110774845440029746985533445760661340160797954694912077059616512004922382751536186668271768031773322854132993274777373
1367108864218103808032262586368287435653120171978391552072929083392002253701971968051357681254400862560447344641053984087808009103522427238452612154522880182299439875482859599056870439620714970178
1426843545693952409601507097313281467838627840969214106009645873784422400160418150154240420831186124800831828704501760123230813040640013463090623554561051447205954560554739876415168176937691643840257363915118315857750536586135
Technical details
Node id
n-4241024ae5c1ee16 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-b056bec77200b8b3

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:      1;
  1:      4,      10;
  2:     16,      80,       91;
  3:     64,     480,     1128,       820;
  4:    256,    2560,     9168,     13840,       7381;
  5:   1024,   12800,    61696,    142720,     157772,      66430;
  6:   4096,   61440,   372480,   1164800,    1975776,    1718880,     597871;
  7:  16384,  286720,  2095104,   8268800,   18995840,   25348800,   18160336,    5380840;
  8:  65536, 1310720, 11210752,  53473280,  155319040,  280870400,  308347360,  187656800,   48427561;
  9: 262144, 5898240, 57802752, 323420160, 1137153024, 2602252800, 3871130880, 3605961600, 1906835220, 435848050;
%F Formula
The row polynomials satisfy (n+2*n*x+n*x^2)*P(n,x) = (4*n+(23*n-5)*x+(42*n-18)*x^2+(31*n-21)*x^3+(8*n-8)*x^4)*P(n-1,x) - (2*x+14*x^2+30*x^3+26*x^4+8*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n + (23*n - 5)*x + (42*n - 18)*x^2 + (31*n - 21)*x^3 + (8*n - 8)*x^4)*P[n - 1, x] - (2*x + 14*x^2 + 30*x^3 + 26*x^4 + 8*x^5)*D[P[n - 1, x], x])/(n + 2*n*x + n*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((4*n + (23*n - 5)*x + (42*n - 18)*x^2 + (31*n - 21)*x^3 + (8*n - 8)*x^4)*v[i - 1] - (2*x + 14*x^2 + 30*x^3 + 26*x^4 + 8*x^5)*deriv(v[i - 1], x))/(n + 2*n*x + n*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

References

  1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  3. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023