Real-rooted atlas

A028326 → basis A013610 → basis A008288

nonnegative coefficients

Proved for all n all roots real, in [−3−2√2, −3+2√2]coefficients of one signHadamard kernel

Recurrence

\(P_{n}(x) = \left(-10 + 3\,x\right)\,P_{n-2}(x) + \left(7 + 3\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 2\)
\(P_{1}(x) = 10 + 6\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
022
110616
2507818146
3250666378541348
4125047824698156616212458
562503131445450256505994486115144
6312501938783794583143341219862187014581064234
715625011565062877282321856218139145321707727443749836332

All 15 rows

Technical details
Node id
n-7779a9e47344358a · revision 0 · JSON
Definition source
derived:basis_map_A008288
Fingerprint
f6-0fee5d4c7376c95e

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:       2;
  1:      10,        6;
  2:      50,       78,        18;
  3:     250,      666,       378,        54;
  4:    1250,     4782,      4698,      1566,       162;
  5:    6250,    31314,     45450,     25650,      5994,       486;
  6:   31250,   193878,    379458,    314334,    121986,     21870,     1458;
  7:  156250,  1156506,   2877282,   3218562,   1813914,    532170,    77274,    4374;
  8:  781250,  6719262,  20397546,  29156814,  22076226,   9314190,  2188458,  266814,  13122;
  9: 3906250, 38282274, 137637306, 241736562, 233520570, 131548050, 44085546, 8621154, 905418, 39366;
%F Formula
The row polynomials satisfy P(n,x) = -(10-3*x)*P(n-2,x) + (7+3*x)*P(n-1,x) for n >= 2, with P(0,x) = 2, P(1,x) = 10 + 6*x.
%t Mathematica
P[0, x_] := 2;
P[1, x_] := 10 + 6*x;
P[n_, x_] := P[n, x] = Expand[-(10 - 3*x)*P[n - 2, x] + (7 + 3*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] = 2; v[2] = 10 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = -(10 - 3*x)*v[i - 2] + (7 + 3*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

  • Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in [−3−2√2, −3+2√2] (rational outer bound [-1564556493/268435456, -46056243/268435456])
    by Affine-current basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
    derived from Affine-current basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
    derived from Affine-current basis map theorem

Proof chains ? JSON

Same fingerprint ?

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.