Real-rooted atlas

A091042 → basis A013610 → basis A013609

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-9 - 36\,x - 36\,x^{2}\right)\,P_{n-2}(x) + \left(10 + 12\,x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 13 + 18\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2131831
3121300180601
4109338224284151210711
5984144280719284924811664186961
68857348714610351441050192498960855363245551
77971615181540136014841829692813124592465004860652856258281
8717445353808390168151140281932920271246320147456288408240004199040974792551

All 15 rows

Technical details
Node id
n-2b8ae16f62b5e385 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-78665ffd6c0972be

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,         18;
   3:       121,        300,         180;
   4:      1093,       3822,        4284,        1512;
   5:      9841,      44280,       71928,       49248,       11664;
   6:     88573,     487146,     1035144,     1050192,      498960,       85536;
   7:    797161,    5181540,    13601484,    18296928,    13124592,     4650048,      606528;
   8:   7174453,   53808390,   168151140,   281932920,   271246320,   147456288,    40824000,     4199040;
   9:  64570081,  548845680,  1989565488,  3996281664,  4829194080,  3556493568,  1532369664,   342641664,   28553472;
  10: 581130733, 5520741954, 22773060432, 53309663136, 77603077728, 72273705408, 42560909568, 14998970880, 2776405248, 191476224;
%F Formula
The row polynomials satisfy P(n,x) = -(9+36*x+36*x^2)*P(n-2,x) + (10+12*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 13 + 18*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 13 + 18*x;
P[n_, x_] := P[n, x] = Expand[-(9 + 36*x + 36*x^2)*P[n - 2, x] + (10 + 12*x)*P[n - 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] = 13 + 18*x; for(n = 3, N, my(i = n); v[i] = -(9 + 36*x + 36*x^2)*v[i - 2] + (10 + 12*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, 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