Real-rooted atlas

A094816 → basis A013611 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(2 + 6\,x + 6\,x^{2} + 2\,x^{3} - n - 3\,n x - 3\,n x^{2} - n x^{3}\right)\,P_{n-2}(x) + \left(4\,x + 4\,x^{2} + n + 2\,n x + n x^{2}\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2 + 6\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2268
35304176
416140376316848
56572428224628272110960
6326430621484511565857425946161792
71957294581760795394368987397755862718012693056
81370022940015795965837856125689241582405610818644310593649978112
910960120090961553990066628088173936614284025016284028540159361480384745611024112896
10986410195709741671095768100278962465565516490096467663763501845245476120247956220251379636622979409920
1198641012100221821954760709105194976403636704490684607357188134420185490144242497896100186291473407183976387360674441560596593664
121085051122462653860247451040801458719628045621891922721490739768712278032149036836526415309523316646817752198441746632470475090616011263282739614777528225792
1313020613453133183286033732010774621559728321729135314057359270903628708405774673405039689275212751928994573084566557794175470071640818037334498128351549491481833790646881418659596652544
1416926797486429926795242492910749698033881483691948156157769118602510568851831502122019292573989621608459000693202838529131487586273417188643645418785213154171408723382160365562455456594095743165663771510612687785738043392
15236975164805632934365871476888505711755564670014375028280691518914320510007809861095478264298860108543235258218652169154479269200989489079901849211980535427627702045604612146535609437328820193694232034023994925681523350666577636838177561409614416067543040
Technical details
Node id
n-9c84183b4d2739be · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-8d3b69a18d4677d2

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,        6;
   3:      5,       30,        41;
   4:     16,      140,       376,       316;
   5:     65,      724,      2822,      4628,       2721;
   6:    326,     4306,     21484,     51156,      58574,      25946;
   7:   1957,    29458,    176079,    539436,     898739,     775586,     271801;
   8:  13700,   229400,   1579596,   5837856,   12568924,   15824056,   10818644,    3105936;
   9: 109601,  2009096,  15539900,  66628088,  173936614,  284025016,  284028540,  159361480,   38474561;
  10: 986410, 19570974, 167109576, 810027896, 2465565516, 4900964676, 6376350184, 5245476120, 2479562202, 513796366;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = -(n-2+(3*n-6)*x+(3*n-6)*x^2+(n-2)*x^3)*P(n-2,x) + (n+(2*n+4)*x+(n+4)*x^2)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 6*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2 + 6*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n - 2 + (3*n - 6)*x + (3*n - 6)*x^2 + (n - 2)*x^3)*P[n - 2, x] + (n + (2*n + 4)*x + (n + 4)*x^2)*P[n - 1, x])/(1 + x)]];
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 + 6*x; for(n = 3, N, my(i = n); v[i] = (-(n - 2 + (3*n - 6)*x + (3*n - 6)*x^2 + (n - 2)*x^3)*v[i - 2] + (n + (2*n + 4)*x + (n + 4)*x^2)*v[i - 1])/(1 + x)); 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