Real-rooted atlas

A122542 → θ+1 → reverse

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
111
222
3347
4412420
552424457
664072364158
7760160152484431
88843004402646041160
9911250410209604087243089
10101447842044270018005848448154
11111801152369663846012304079296421367
12122201620619213328165481182047601032108455644
13132642200978025344395363735221180704013041204144137
141431229041474044820848161011367585235340996016081324371638
151536437442138474800167220243072230512141792557401360019441444954335
Technical details
Node id
n-b1e35c21e30f7979 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-37326d697e2d9020

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
  • 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:  1;
   2:  2;
   3:  3,   4;
   4:  4,  12,   4;
   5:  5,  24,  24,    4;
   6:  6,  40,  72,   36,    4;
   7:  7,  60, 160,  152,   48,    4;
   8:  8,  84, 300,  440,  264,   60,   4;
   9:  9, 112, 504, 1020,  960,  408,  72,  4;
  10: 10, 144, 784, 2044, 2700, 1800, 584, 84, 4;
%F Formula
The row polynomials satisfy P(n,x) = -x^2*P(n-4,x) - (2*x+2*x^2)*P(n-3,x) - (1+x^2)*P(n-2,x) + (2+2*x)*P(n-1,x) for n >= 5, with P(1,x) = 1, P(2,x) = 2, P(3,x) = 3 + 4*x, P(4,x) = 4 + 12*x + 4*x^2.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2;
P[3, x_] := 3 + 4*x;
P[4, x_] := 4 + 12*x + 4*x^2;
P[n_, x_] := P[n, x] = Expand[-x^2*P[n - 4, x] - (2*x + 2*x^2)*P[n - 3, x] - (1 + x^2)*P[n - 2, x] + (2 + 2*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2; v[3] = 3 + 4*x; v[4] = 4 + 12*x + 4*x^2; for(n = 5, N, my(i = n); v[i] = -x^2*v[i - 4] - (2*x + 2*x^2)*v[i - 3] - (1 + x^2)*v[i - 2] + (2 + 2*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, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.