Real-rooted atlas

A111125 → basis A013611 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1\right)\,P_{n-2}(x) + \left(7 + 4\,x\right)\,P_{n-1}(x) + \left(4\,x\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 8 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
28412
3557616147
43771052544642037
525841300412352339225631588
617711152328236752113792194561024541063
7121393173344841482403105664902400105472409610120713
883204019401172689206407548678433158400645632054886416384204820604
95702887215007604110726604817068257921063687936306848768428359682768896655364451009435
1039088169236849402017400100496367832674563125896729612533212160255476121626823884813631488262144103220023293
11267914296259947792922693682174727667756578568650256296964631257487361292876103681960463564816045178886579814410485762541121557972
1218363119032846457353764127102278752156094271016962293143314713615951964522496585786415513612030444503041409811087369247129600312475648419430466117862611087
13125862690253112453557456627757489053123123207483893765889956451612165219656489472002452723592888326551840774553610308836524032961485209600516738252801463812096167772161811297074412177
148626757127234002688472740949958945530752616958778791769614773251399118336164331031376967689685447869587456327303066379878466397727431065682569962979328627429815091228134552371267779952646710886452071578485762540
1559128672987937126218992777214322619187946928120709683004169856363874137686231296502264824665338880365736024148709376153232346303561728389769275167866886205456020144128625120191184896394443505008641498155057152310714040322684354561566359966046128931
Technical details
Node id
n-300f32295f9afdcb · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-d0568900822197f8

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:        8,          4;
   3:       55,         76,          16;
   4:      377,       1052,         544,          64;
   5:     2584,      13004,       12352,        3392,         256;
   6:    17711,     152328,      236752,      113792,       19456,        1024;
   7:   121393,    1733448,     4148240,     3105664,      902400,      105472,       4096;
   8:   832040,   19401172,    68920640,    75486784,    33158400,     6456320,     548864,     16384;
   9:  5702887,  215007604,  1107266048,  1706825792,  1063687936,   306848768,   42835968,   2768896,    65536;
  10: 39088169, 2368494020, 17400100496, 36783267456, 31258967296, 12533212160, 2554761216, 268238848, 13631488, 262144;
%F Formula
The row polynomials satisfy P(n,x) = -P(n-2,x) + (7+4*x)*P(n-1,x) + 4*x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 8 + 4*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 8 + 4*x;
P[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (7 + 4*x)*P[n - 1, x] + 4*x*D[P[n - 1, x], 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] = 8 + 4*x; for(n = 3, N, my(i = n); v[i] = -v[i - 2] + (7 + 4*x)*v[i - 1] + 4*x*deriv(v[i - 1], x)); 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

  • Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
    by Ray basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
    derived from Ray basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
    derived from Ray basis map theorem

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023