Real-rooted atlas

A091042 → basis A013610 → basis A007318

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(-3 + 2\,n\right)\,P_{n}(x) = \left(-39 - 27\,x + 26\,n + 18\,n x\right)\,P_{n-1}(x) + \left(-16 - 28\,x - 12\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
213922
312115045316
41093191110711894264
598412214017982615672956848
688573243573258786131274311852673756064
7797161259077034003712287116820287145314947710050496
8717445326904195420377853524161516952895460800963787532805133589632
9645700812744228404973913724995352083018246301111404242394327626768881115371775616256
10581130733276037097756932651086663707892485019235822585532946650142121171794601084533337397723600633344
1152301766012745842715063497612745848780024407229959317040757763060152690230983706545096546852789427121101240029313687948288
1247071589413270661639119693570450195104247021910510174260870906737713765023069459909189532773894619515832281245504090716433336740743814169384372224
13423644304721264777690450074468725438501243213999650013679103288375104101557210005590330886700211993214580055806406177598058737700106719257706200145001328602555417383817216
1438127987424932573639151182178817699004886144700180400574177220440083295152416333012167943170097138924229438690264413646177782731309764174791547375945674245141661422230060808943046721736580303675392
153431518868244124878511794769082410070320164716501450401367562226493180809369213568144997319014968603846582477754491805479922967337660824998284641311208616439763888405221376992101218652888226784148367941387061019790259041386496

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(P_{n}(x) = \left(-9 - 18\,x - 9\,x^{2}\right)\,P_{n-2}(x) + \left(10 + 6\,x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 13 + 9\,x\)
Technical details
Node id
n-913df248d683398b · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-4e7c1404f4ed504f)
Fingerprint
f6-fb43674541c36216

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:        13,          9;
   3:       121,        150,         45;
   4:      1093,       1911,       1071,        189;
   5:      9841,      22140,      17982,       6156,        729;
   6:     88573,     243573,     258786,     131274,      31185,       2673;
   7:    797161,    2590770,    3400371,    2287116,     820287,     145314,      9477;
   8:   7174453,   26904195,   42037785,   35241615,   16952895,    4608009,    637875,     32805;
   9:  64570081,  274422840,  497391372,  499535208,  301824630,  111140424,  23943276,   2676888,   111537;
  10: 581130733, 2760370977, 5693265108, 6663707892, 4850192358, 2258553294, 665014212, 117179460, 10845333, 373977;
%F Formula
The row polynomials satisfy (2*n-3)*P(n,x) = (26*n-39+(18*n-27)*x)*P(n-1,x) - (16+28*x+12*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((26*n - 39 + (18*n - 27)*x)*P[n - 1, x] - (16 + 28*x + 12*x^2)*D[P[n - 1, x], x])/(2*n - 3)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((26*n - 39 + (18*n - 27)*x)*v[i - 1] - (16 + 28*x + 12*x^2)*deriv(v[i - 1], x))/(2*n - 3)); 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