Real-rooted atlas

A030528 → basis A013609 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(3 + 2\,x\right)\,P_{n-2}(x) + \left(2\,x\right)\,P_{n-2}'(x) + \left(3 + 2\,x\right)\,P_{n-1}(x) + \left(2\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1426
21520439
357148728285
4216984868224162308
58196206884439686403220509
6310538020823645715215600172864198033
7117722289787269087324963008805536044801282061002
844631136474461989008725376506057613751041823361126425622963187
9169209808141651668100989724647789176029318944566316856729627648512272360517
1064152047658480423961320108501456011278713765649272961490706562152652816872966656010243422426616
11243218728032093834408857321160714205615631368544101367273923509499840685062656767715844838400157696204845375209073
12922112116460445042770988820012195910364820973595235217263999078476281866816193291571202903897088259921920134604803686404096632488876769
13349599249655133826221908148408126439775920027461730212642826831978080156242393619249982446950496661144320115213040648427386883650355285196881929240311957182
141325431355659425374017699286129161297644783884835287712435472449049535424643057953098284812103023258624293004304000044653582848043255032832263420313696858112194969616384141100890396687
Technical details
Node id
n-22eddc5a9752c540 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-06541b61383eaca3

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:      4,       2;
  2:     15,      20,        4;
  3:     57,     148,       72,        8;
  4:    216,     984,      868,      224,       16;
  5:    819,    6206,     8844,     3968,      640,       32;
  6:   3105,   38020,    82364,    57152,    15600,     1728,      64;
  7:  11772,  228978,   726908,   732496,   300880,    55360,    4480,    128;
  8:  44631, 1364744,  6198900,  8725376,  5060576,  1375104,  182336,  11264,   256;
  9: 169209, 8081416, 51668100, 98972464, 77891760, 29318944, 5663168, 567296, 27648, 512;
%F Formula
The row polynomials satisfy P(n,x) = (3+2*x)*P(n-2,x) + 2*x*d/dx P(n-2,x) + (3+2*x)*P(n-1,x) + 2*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 2*x;
P[n_, x_] := P[n, x] = Expand[(3 + 2*x)*P[n - 2, x] + 2*x*D[P[n - 2, x], x] + (3 + 2*x)*P[n - 1, x] + 2*x*D[P[n - 1, x], 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] = 4 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (3 + 2*x)*v[i - 2] + 2*x*deriv(v[i - 2], x) + (3 + 2*x)*v[i - 1] + 2*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 - 1, v[i])))};
T(9)

polytool c48629374597

Properties

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

Proof chains ? JSON

References

  1. [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
  2. [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