Real-rooted atlas

A207607 → basis A013609 → basis A007318

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1\right)\,P_{n-2}(x) + \left(5 + 2\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 7 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
17411
23434876
316323410816521
478114621000296323571
537428638781634647526424476
617929492125535632656106561824128167761
7859032732803673882705281178403036842882561149851

All 15 rows

Technical details
Node id
n-3bbb6ca3110db0e1 · revision 0 · JSON
Definition source
derived:basis_map_A007318
Fingerprint
f6-e30221224ac617c7

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:       7,       4;
  2:      34,      34,        8;
  3:     163,     234,      108,       16;
  4:     781,    1462,     1000,      296,      32;
  5:    3742,    8638,     7816,     3464,     752,      64;
  6:   17929,   49212,    55356,    32656,   10656,    1824,     128;
  7:   85903,  273280,   367388,   270528,  117840,   30368,    4288,    256;
  8:  411586, 1488994,  2328144,  2054760, 1119600,  385696,   82048,   9856,   512;
  9: 1972027, 7994862, 14251320, 14659560, 9589680, 4137312, 1177344, 213120, 22272, 1024;
%F Formula
The row polynomials satisfy P(n,x) = -P(n-2,x) + (5+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 7 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 7 + 4*x;
P[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (5 + 2*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] = 7 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -v[i - 2] + (5 + 2*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