Real-rooted atlas

A166979 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-10 - 2\,x - x^{2}\right)\,P_{n-2}(x) + \left(-9\,x - 9\,x^{2}\right)\,P_{n-2}'(x) + \left(11 + 2\,x\right)\,P_{n-1}(x) + \left(9\,x\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 11 + 2\,x\)
\(P_{2}(x) = 111 + 60\,x + 3\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8Pn(1)
111213
2111603174
31111136217442651
411111281006915380546511
5111111556122235770232807056926994
611111111078854074357431175720615301176720573827
71111111120720448222438361453415964430003013851618208500555545
8111111111395910738065935479152275568820264571335127784162780822664913216965732

All 15 rows

Technical details
Node id
n-500e32224f9dbdfe · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-dd3f1b3bb23145ea

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:          11,             2;
   2:         111,            60,             3;
   3:        1111,          1362,           174,             4;
   4:       11111,         28100,          6915,           380,            5;
   5:      111111,        556122,        235770,         23280,          705,           6;
   6:     1111111,      10788540,       7435743,       1175720,        61530,        1176,          7;
   7:    11111111,     207204482,     224383614,      53415964,      4300030,      138516,       1820,        8;
   8:   111111111,    3959107380,    6593547915,    2275568820,    264571335,    12778416,     278082,     2664,      9;
   9:  1111111111,   75445262442,  190558002690,   92987443560,  15014742435,  1020298482,   32705400,   512280,   3735,   10;
  10: 11111111111, 1435682208620, 5448791968983, 3694612748640, 806912456460, 74134107432, 3283493829, 74774040, 882585, 5060, 11;
%F Formula
The row polynomials satisfy P(n,x) = -(10+2*x+x^2)*P(n-2,x) - (9*x+9*x^2)*d/dx P(n-2,x) + (11+2*x)*P(n-1,x) + 9*x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 11 + 2*x, P(2,x) = 111 + 60*x + 3*x^2.
%t Mathematica
P[1, x_] := 11 + 2*x;
P[2, x_] := 111 + 60*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[-(10 + 2*x + x^2)*P[n - 2, x] - (9*x + 9*x^2)*D[P[n - 2, x], x] + (11 + 2*x)*P[n - 1, x] + 9*x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 11 + 2*x; v[2] = 111 + 60*x + 3*x^2; for(n = 3, N, my(i = n); v[i] = -(10 + 2*x + x^2)*v[i - 2] - (9*x + 9*x^2)*deriv(v[i - 2], x) + (11 + 2*x)*v[i - 1] + 9*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, v[i])))};
T(10)

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.