Real-rooted atlas

A111125 → basis A007318 → basis A013609

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2527
31918441
471108528239
5265556420136161393
6989264627401376336328119
7369112006158321084840808006447321
81377552760846007368037680113281856128275807

All 15 rows

Technical details
Node id
n-243f9a1a4ce1746b · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-dbf3a85c57f4456b

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:      1;
   2:      5,      2;
   3:     19,     18,       4;
   4:     71,    108,      52,       8;
   5:    265,    556,     420,     136,      16;
   6:    989,   2646,    2740,    1376,     336,      32;
   7:   3691,  12006,   15832,   10848,    4080,     800,     64;
   8:  13775,  52760,   84600,   73680,   37680,   11328,   1856,   128;
   9:  51409, 226584,  428088,  453072,  294000,  119872,  30016,  4224,  256;
  10: 191861, 956394, 2080920, 2594784, 2044464, 1056160, 357952, 76800, 9472, 512;
%F Formula
The row polynomials satisfy P(n,x) = -P(n-2,x) + (4+2*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 5 + 2*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 5 + 2*x;
P[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (4 + 2*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 5 + 2*x; for(n = 3, N, my(i = n); v[i] = -v[i - 2] + (4 + 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, v[i])))};
T(10)

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