Real-rooted atlas

A078812 → basis A013609 → basis A013609

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2549
324401680
411529224064711
55511880235212802566319
626401131219040157446400102456160
7126496524013809615360094720307204096499121
86060536548493240013046401081600531456143360163844435929

All 15 rows

Technical details
Node id
n-5d25926d30ecf6f4 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-4a3147452b8a179e

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,        4;
   3:      24,       40,       16;
   4:     115,      292,      240,       64;
   5:     551,     1880,     2352,     1280,      256;
   6:    2640,    11312,    19040,    15744,     6400,     1024;
   7:   12649,    65240,   138096,   153600,    94720,    30720,     4096;
   8:   60605,   365484,   932400,  1304640,  1081600,   531456,   143360,    16384;
   9:  290376,  2004600,  5985840, 10099200, 10531840,  6952960,  2838528,   655360,   65536;
  10: 1391275, 10819020, 37015200, 73134720, 91974400, 76360704, 41861120, 14614528, 2949120, 262144;
%F Formula
The row polynomials satisfy P(n,x) = -P(n-2,x) + (5+4*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 5 + 4*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 5 + 4*x;
P[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (5 + 4*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 + 4*x; for(n = 3, N, my(i = n); v[i] = -v[i - 2] + (5 + 4*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