Real-rooted atlas

A232535 → basis A007318 → basis A007318

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1 - 2\,x - x^{2}\right)\,P_{n-2}(x) + \left(6 + 2\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 5 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1527
22920352
3169166494388
498512565549652896
55741900252861416165621616
63346162268456691660830712607161344
719502542004636951916999643359595038581204288

All 15 rows

Technical details
Node id
n-86243365a11d35b7 · revision 0 · JSON
Definition source
derived:basis_map_A007318
Fingerprint
f6-12c19fdbc0f3a4d9

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:       5,        2;
  2:      29,       20,        3;
  3:     169,      166,       49,        4;
  4:     985,     1256,      554,       96,       5;
  5:    5741,     9002,     5286,     1416,     165,       6;
  6:   33461,    62268,    45669,    16608,    3071,     260,      7;
  7:  195025,   420046,   369519,   169996,   43359,    5950,    385,     8;
  8: 1136689,  2781136,  2853540,  1588800,  518190,   99408,  10612,   544,   9;
  9: 6625109, 18150098, 21278876, 13910800, 5533870, 1370164, 206844, 17760, 741, 10;
%F Formula
The row polynomials satisfy P(n,x) = -(1+2*x+x^2)*P(n-2,x) + (6+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 5 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(1 + 2*x + x^2)*P[n - 2, x] + (6 + 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] = 5 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(1 + 2*x + x^2)*v[i - 2] + (6 + 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

Candidate equalities ?

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