Real-rooted atlas

A156992 → 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

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
110616
215023454438
330007992453664816176
475000286920281880939609720747480
5225000011273040163620009234000215784017496041451840
6787500004885725609562341607921216803074047205511240036741602681869680
731500000002331516096058006005120648862099203656850624010728547200155784384088179840198300453120

All 15 rows

Technical details
Node id
n-8d3044b53af1a6c6 · 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:            10,              6;
  2:           150,            234,              54;
  3:          3000,           7992,            4536,             648;
  4:         75000,         286920,          281880,           93960,            9720;
  5:       2250000,       11273040,        16362000,         9234000,         2157840,          174960;
  6:      78750000,      488572560,       956234160,       792121680,       307404720,        55112400,        3674160;
  7:    3150000000,    23315160960,     58006005120,     64886209920,     36568506240,     10728547200,     1557843840,       88179840;
  8:  141750000000,  1219142897280,   3700930746240,   5290212332160,   4005510445440,   1689966633600,   397073819520,    48410732160,    2380855680;
  9: 7087500000000, 69459357945600, 249729128006400, 438606818092800, 423699722208000, 238680781920000, 79988814662400, 15642221817600, 1642790419200, 71425670400;
%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+7+(3*n+3)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 10 + 6*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 10 + 6*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 + 7 + (3*n + 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] = 1; v[2] = 10 + 6*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 + 7 + (3*n + 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.