Real-rooted atlas

A172431 → basis A013610 → basis A013611

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-16 - 96\,x - 144\,x^{2}\right)\,P_{n-2}(x) + \left(9 + 24\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 9 + 24\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
192433
265336432833
344133368352691219041
4292928992106704172800103680415105
51930523551211387522723328321408014929928823969
612688118378721098878434656768606735365573836820901888184924097
7833049139645209939888038883456090097920012332113929196830722866544643843559137

All 15 rows

Technical details
Node id
n-23a4146af0ae58e7 · revision 0 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-18c120bb8aec7641

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:       65,       336,        432;
  3:      441,      3336,       8352,        6912;
  4:     2929,     28992,     106704,      172800,       103680;
  5:    19305,    235512,    1138752,     2723328,      3214080,      1492992;
  6:   126881,   1837872,   10988784,    34656768,     60673536,     55738368,     20901888;
  7:   833049,  13964520,   99398880,   388834560,    900979200,   1233211392,    919683072,    286654464;
  8:  5467345, 104087328,  859211280,  4010999040,  11560631040,  21015355392,  23451918336,  14619377664,   3869835264;
  9: 35877321, 764597208, 7180062336, 38945525760, 134252432640, 304375781376, 452591566848, 423961952256, 225740390400, 51597803520;
%F Formula
The row polynomials satisfy P(n,x) = -(16+96*x+144*x^2)*P(n-2,x) + (9+24*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[-(16 + 96*x + 144*x^2)*P[n - 2, x] + (9 + 24*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] = -(16 + 96*x + 144*x^2)*v[i - 2] + (9 + 24*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