Real-rooted atlas

A196347 → 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

\(P_{n}(x) = \left(10\,n - 3\,n x - 10\,n^{2} + 3\,n^{2} x\right)\,P_{n-2}(x) + \left(7\,n + 3\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 5 + 3\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1538
2507818146
3750199811341624044
4150005738456376187921944149496
5375000187884027270001539000359640291606908640
61125000069796080136604880113160240439149607873200524880383124240
7393750000291439512072507506408110776240457106328013410684001947304801102248024787556640

All 15 rows

Technical details
Node id
n-4a08a6e883134505 · 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:            1;
  1:            5,             3;
  2:           50,            78,             18;
  3:          750,          1998,           1134,            162;
  4:        15000,         57384,          56376,          18792,           1944;
  5:       375000,       1878840,        2727000,        1539000,         359640,          29160;
  6:     11250000,      69796080,      136604880,      113160240,       43914960,        7873200,        524880;
  7:    393750000,    2914395120,     7250750640,     8110776240,     4571063280,     1341068400,     194730480,      11022480;
  8:  15750000000,  135460321920,   411214527360,   587801370240,   445056716160,   187774070400,   44119313280,    5378970240,    264539520;
  9: 708750000000, 6945935794560, 24972912800640, 43860681809280, 42369972220800, 23868078192000, 7998881466240, 1564222181760, 164279041920, 7142567040;
%F Formula
The row polynomials satisfy P(n,x) = -(10*n^2-10*n-(3*n^2-3*n)*x)*P(n-2,x) + (7*n+3*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 3*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 5 + 3*x;
P[n_, x_] := P[n, x] = Expand[-(10*n^2 - 10*n - (3*n^2 - 3*n)*x)*P[n - 2, x] + (7*n + 3*n*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] = 5 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = -(10*n^2 - 10*n - (3*n^2 - 3*n)*x)*v[i - 2] + (7*n + 3*n*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.