Real-rooted atlas

A207607 → basis A013610 → basis A013611

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1\right)\,P_{n-2}(x) + \left(6 + 12\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 9 + 24\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
192433
253252288593
330921244752345610641
4180116200537127776041472190945
510497116688511920110764811819524976643426369
6611818098924418064127111682034201617169408597196861483697
7356589547683635715168128176128273404160346622976241864704716636161103280177

All 15 rows

Technical details
Node id
n-3e7e29234c95e761 · revision 0 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-c7a742ab1938f2d8

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:       53,       252,        288;
  3:      309,      2124,       4752,        3456;
  4:     1801,     16200,      53712,       77760,       41472;
  5:    10497,    116688,     511920,     1107648,     1181952,      497664;
  6:    61181,    809892,    4418064,    12711168,    20342016,    17169408,     5971968;
  7:   356589,   5476836,   35715168,   128176128,   273404160,   346622976,   241864704,    71663616;
  8:  2078353,  36330192,  275594976,  1184927616,  3158196480,  5343418368,  5604691968,  3332358144,   859963392;
  9: 12113529, 237444552, 2053816992, 10288529280, 32894906112, 69612244992, 97507307520, 87178788864, 45148078080, 10319560704;
%F Formula
The row polynomials satisfy P(n,x) = -P(n-2,x) + (6+12*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[-P[n - 2, x] + (6 + 12*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] = -v[i - 2] + (6 + 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 - 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