Real-rooted atlas

A091044 → basis A013611 → basis A013609

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-16 - 64\,x - 64\,x^{2}\right)\,P_{n-2}(x) + \left(12 + 16\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 12 + 16\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1121628
2128320192640
3134448645632204813888
414080665601136648192020480296704
51474568601601941504209715210649601966086307840
6154419210715136300810244338483233488896128450561835008133894144
716171008130088960436862976789053440820510720484442112146800640167772162840707072

All 15 rows

Technical details
Node id
n-2de3c606f08cd985 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-ade2b5b96bd8d678

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:         12,          16;
  2:        128,         320,         192;
  3:       1344,        4864,        5632,         2048;
  4:      14080,       66560,      113664,        81920,        20480;
  5:     147456,      860160,     1941504,      2097152,      1064960,       196608;
  6:    1544192,    10715136,    30081024,     43384832,     33488896,     12845056,      1835008;
  7:   16171008,   130088960,   436862976,    789053440,    820510720,    484442112,    146800640,    16777216;
  8:  169345024,  1549533184,  6057885696,  13153337344,  17233346560,  13816037376,   6517948416,  1610612736,   150994944;
  9: 1773404160, 18187550720, 81136713728, 205856440320, 325666734080, 330762813440, 213406187520, 82946555904, 17112760320, 1342177280;
%F Formula
The row polynomials satisfy P(n,x) = -(16+64*x+64*x^2)*P(n-2,x) + (12+16*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 12 + 16*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 12 + 16*x;
P[n_, x_] := P[n, x] = Expand[-(16 + 64*x + 64*x^2)*P[n - 2, x] + (12 + 16*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] = 12 + 16*x; for(n = 2, N, my(i = n + 1); v[i] = -(16 + 64*x + 64*x^2)*v[i - 2] + (12 + 16*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

Same fingerprint ?

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