Real-rooted atlas

A232535 → basis A013610 → basis A013611

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(-9 - 72\,x - 144\,x^{2}\right)\,P_{n-2}(x) + \left(10 + 24\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 9 + 24\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
192433
281384432897
3729492010512691223073
4656157408180000235008103680582657
559049634776262396851425284665600149299214618913
6531441677587234776144910586881263029768559820820901888365945217
747829697054879243255972814185981442658334464279736934414840340482866544649152881953
843046721721032576514139270420268278784479289830406879707136057438388224246522839043869835264228860312577
9387420489726413416859057410752272005502976777375843840141741176832016279465328641112792629248396013142016515978035205721852188193
10348678440172350776320660527719920348100941312011670853040640258424453693443792523957862435934430298112205862336593926191736422400681091006464143049404068737
11313810596097139191061047231372923600429165409881601651677103641604306283675688967708237970227209382154742005767501791754321923667675069808649467164989849689161004482563576262995993633
1228242953648169901310279047779168545443251341276770560022308877405785606696159941689344141607327312281602110658661502156821785173551415296149594828659752966332784178102272142063200475545611590930582732889406825948317697
13254186582832967994910907800824676594678432599141605729075229012965785296640985551928341442562405883589822218244249697141163294725391080607820677124800221574148915202870703239504855041064820156200386562097958435474636814979048753070082235172908144234273
1422876792454961657707783080128863455935221928068478103715102208365677931582496000138704231184209510438403902728156815367850038688949731328118354674350714388481299629871885975552010120092202105307136533159624944543334417501146086868254723055725945626296321925877696823296055879343038532483457
Technical details
Node id
n-247f017b86539b03 · revision 0 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-5ab10b49b3d45946

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:         9,         24;
  2:        81,        384,         432;
  3:       729,       4920,       10512,         6912;
  4:      6561,      57408,      180000,       235008,       103680;
  5:     59049,     634776,     2623968,      5142528,      4665600,       1492992;
  6:    531441,    6775872,    34776144,     91058688,    126302976,      85598208,      20901888;
  7:   4782969,   70548792,   432559728,   1418598144,   2658334464,    2797369344,    1484034048,     286654464;
  8:  43046721,  721032576,  5141392704,  20268278784,  47928983040,   68797071360,   57438388224,   24652283904,   3869835264;
  9: 387420489, 7264134168, 59057410752, 272005502976, 777375843840, 1417411768320, 1627946532864, 1112792629248, 396013142016, 51597803520;
%F Formula
The row polynomials satisfy P(n,x) = -(9+72*x+144*x^2)*P(n-2,x) + (10+24*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 9 + 24*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 9 + 24*x;
P[n_, x_] := P[n, x] = Expand[-(9 + 72*x + 144*x^2)*P[n - 2, x] + (10 + 24*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 9 + 24*x; for(n = 2, N, my(i = n + 1); v[i] = -(9 + 72*x + 144*x^2)*v[i - 2] + (10 + 24*x)*v[i - 1]); 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