Real-rooted atlas

A113953 → basis A013609 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(6 + 4\,x\right)\,P_{n-2}(x) + \left(4\,x\right)\,P_{n-2}'(x) + \left(3 + 2\,x\right)\,P_{n-1}(x) + \left(2\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1325
21520439
363162768309
42791196992240162723
5121584101104845926883225985
65319575561128287171218064185664267399
72324738737810932209979123756326384048001282946157
810165525802201023711212909776681203217016962090241203225634563803
9444447170711789367481615899750411300745638908480693811264614429440512429718089
101943271112453604843506516189165158417625808328033173761949661442610585619095047065610245638506367
11849649573864413075091849562194853162426293846368154319378564894904192883453184921955845442048166912204877806805397
12371491114842522764663004885922499789091043794727579362811394599681133496924162647597619236956070403092797441505280038912040961125617288883
131624263033170733932258187163157628075214681765340416902096492016426518424710991833607264080051201302057310721448343705699422003240603648897024819217025076118161
14710173575207435373300508409303102031196258794976737079451704168347048624249651431864893120186447164139524173061872896592335817728537634826243083198464107204608204800016384268565863733559
Technical details
Node id
n-2acd3d9f16b4d004 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-369971fbfbadebbb

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:      3,        2;
  2:     15,       20,        4;
  3:     63,      162,       76,         8;
  4:    279,     1196,      992,       240,        16;
  5:   1215,     8410,    11048,      4592,       688,       32;
  6:   5319,    57556,   112828,     71712,     18064,     1856,      64;
  7:  23247,   387378,  1093220,    997912,    375632,    63840,    4800,    128;
  8: 101655,  2580220, 10237112,  12909776,   6812032,  1701696,  209024,  12032,   256;
  9: 444447, 17071178, 93674816, 158997504, 113007456, 38908480, 6938112, 646144, 29440, 512;
%F Formula
The row polynomials satisfy P(n,x) = (6+4*x)*P(n-2,x) + 4*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) = 3 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[(6 + 4*x)*P[n - 2, x] + 4*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] = 3 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (6 + 4*x)*v[i - 2] + 4*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