Real-rooted atlas

A086645 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(-2 - 4\,x - 2\,x^{2} + n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(-8 - 46\,x - 84\,x^{2} - 62\,x^{3} - 16\,x^{4} + 4\,n + 23\,n x + 42\,n x^{2} + 31\,n x^{3} + 8\,n x^{4}\right)\,P_{n-1}(x) + \left(-2\,x - 14\,x^{2} - 30\,x^{3} - 26\,x^{4} - 8\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2 + 5\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2257
38404189
4322405463651183
5128128045126560328115761
6512640030560692007381029525210007
72048307201850885708809466327971602657212798249
88192143360104294440768009220288120321608370194239148537285423
932768655360558694426460160759833601351936001452826888609344021523361496811681
101310722949120288276481604198405591485441263916800185008416016889112008716960981937102456619794727
11524288131072001445068809240576003790510080104090112001935565440024039744000190683522008716961000174339220188205821049
122097152576716807079198725114757120241431674887809736704017648039270427834026880030002009136021035116520086297913922156905298051175303342143
1383886082516582403403677696274202624001464196792325455046246401452780650496278448390144038092821638403625247878400227708813788884728860944014121476824115660394400561
1433554432109051904016112418816143198781440853510062080360018247680011048880340992249628658073604149364413542450088750732800426606505133122426687532984082610639420661270932914165208667791528567
151342177284697620480752793681927316543897604815394963456226890481664007887956901888020543072501760040252820876083259023108917248063724656990668849104677554688025521921582567280068773592360114383962274812780405531794889
Technical details
Node id
n-2aa40b2216359e88 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-5b4d63586210805c

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:      1;
   2:      2,       5;
   3:      8,      40,       41;
   4:     32,     240,      546,       365;
   5:    128,    1280,     4512,      6560,      3281;
   6:    512,    6400,    30560,     69200,     73810,      29525;
   7:   2048,   30720,   185088,    570880,    946632,     797160,     265721;
   8:   8192,  143360,  1042944,   4076800,   9220288,   12032160,    8370194,    2391485;
   9:  32768,  655360,  5586944,  26460160,  75983360,  135193600,  145282688,   86093440,  21523361;
  10: 131072, 2949120, 28827648, 160419840, 559148544, 1263916800, 1850084160, 1688911200, 871696098, 193710245;
%F Formula
The row polynomials satisfy (n-2+(2*n-4)*x+(n-2)*x^2)*P(n,x) = (4*n-8+(23*n-46)*x+(42*n-84)*x^2+(31*n-62)*x^3+(8*n-16)*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 >= 3, with P(1,x) = 1, P(2,x) = 2 + 5*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2 + 5*x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n - 8 + (23*n - 46)*x + (42*n - 84)*x^2 + (31*n - 62)*x^3 + (8*n - 16)*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 + (2*n - 4)*x + (n - 2)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 5*x; for(n = 3, N, my(i = n); v[i] = ((4*n - 8 + (23*n - 46)*x + (42*n - 84)*x^2 + (31*n - 62)*x^3 + (8*n - 16)*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 + (2*n - 4)*x + (n - 2)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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