Real-rooted atlas

A207607 → basis A013609 → basis A013611

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1\right)\,P_{n-2}(x) + \left(5 + 8\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 7 + 16\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
171623
234136128298
3163936172810243851
478158481600018944819249765
537423455212505622169619251265536643094
6179291968488856962089984272793618677765242888310457
78590310931205878208173137923016704031096832175636484194304107392847

All 15 rows

Technical details
Node id
n-414e22f737101a4e · revision 0 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-6af366c4ed68c659

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:       7,       16;
  2:      34,      136,       128;
  3:     163,      936,      1728,      1024;
  4:     781,     5848,     16000,     18944,       8192;
  5:    3742,    34552,    125056,    221696,     192512,      65536;
  6:   17929,   196848,    885696,   2089984,    2727936,    1867776,     524288;
  7:   85903,  1093120,   5878208,  17313792,   30167040,   31096832,   17563648,    4194304;
  8:  411586,  5955976,  37250304, 131504640,  286617600,  394952704,  336068608,  161480704,   33554432;
  9: 1972027, 31979448, 228021120, 938211840, 2454958080, 4236607488, 4822401024, 3491758080, 1459617792, 268435456;
%F Formula
The row polynomials satisfy P(n,x) = -P(n-2,x) + (5+8*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 7 + 16*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 7 + 16*x;
P[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (5 + 8*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] = 7 + 16*x; for(n = 2, N, my(i = n + 1); v[i] = -v[i - 2] + (5 + 8*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