Real-rooted atlas

A225470 → basis A013609 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-9 + 12\,x - 4\,x^{2} + 18\,n - 12\,n x - 9\,n^{2}\right)\,P_{n-2}(x) + \left(-3 + 4\,x + 6\,n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
122
218826
319814424366
427002520720646004
54438848600183602880160114388
68573041052352476280100800100803842497200
719053144255286081315893633868804636803225689661624400

All 15 rows

Technical details
Node id
n-902599b8e0aaf369 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-f78dd03c19e9c793

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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:           0;
  1:           2;
  2:          18,           8;
  3:         198,         144,          24;
  4:        2700,        2520,         720,         64;
  5:       44388,       48600,       18360,       2880,       160;
  6:      857304,     1052352,      476280,     100800,     10080,      384;
  7:    19053144,    25528608,    13158936,    3386880,    463680,    32256,     896;
  8:   479320416,   688899168,   392400288,  116375616,  19595520,  1886976,   96768,   2048;
  9: 13470730272, 20517559200, 12661505280, 4188542400, 820018080, 97977600, 7015680, 276480, 4608;
%F Formula
The row polynomials satisfy P(n,x) = -(9*n^2-18*n+9+(12*n-12)*x+4*x^2)*P(n-2,x) + (6*n-3+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[-(9*n^2 - 18*n + 9 + (12*n - 12)*x + 4*x^2)*P[n - 2, x] + (6*n - 3 + 4*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] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 - 18*n + 9 + (12*n - 12)*x + 4*x^2)*v[i - 2] + (6*n - 3 + 4*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

Proof chains ? JSON

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.