Real-rooted atlas

A008277 → θ+2 (shift +1) → D+1

nonnegative coefficients

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence
\(P_{n}(x) = \left(-2 + x + x^{2}\right)\,P_{n-4}(x) + \left(-x + x^{2}\right)\,P_{n-4}'(x) + \left(-1 - 3\,x + x^{3}\right)\,P_{n-3}(x) + \left(-x - x^{2} + x^{3}\right)\,P_{n-3}'(x) + \left(3 - x - 3\,x^{2}\right)\,P_{n-2}(x) + \left(x - 2\,x^{2}\right)\,P_{n-2}'(x) + \left(1 + 3\,x\right)\,P_{n-1}(x) + \left(x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 2\)
\(P_{1}(x) = 5 + 3\,x\)
\(P_{2}(x) = 11 + 17\,x + 4\,x^{2}\)
\(P_{3}(x) = 23 + 69\,x + 39\,x^{2} + 5\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
022
1538
21117432
32369395136
447245250746622
595813133568512573060
61912597645451101575195816130
738381092937933705156103206287990688

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-b7bbaef8a6498cf0 · revision 1 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-50f098c3ca059732

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

OEIS matches ?

No OEIS matches.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:    2;
  1:    5,     3;
  2:   11,    17,      4;
  3:   23,    69,     39,       5;
  4:   47,   245,    250,      74,       6;
  5:   95,   813,   1335,     685,     125,      7;
  6:  191,  2597,   6454,    5110,    1575,    195,     8;
  7:  383,  8109,  29379,   33705,   15610,   3206,   287,     9;
  8:  767, 24965, 128650,  205674,  134316,  40698,  5964,   404,  10;
  9: 1535, 76173, 548895, 1191125, 1054095, 442029, 94290, 10350, 549, 11;
%F Formula
The row polynomials satisfy P(n,x) = -(2-x-x^2)*P(n-4,x) - (x-x^2)*d/dx P(n-4,x) - (1+3*x-x^3)*P(n-3,x) - (x+x^2-x^3)*d/dx P(n-3,x) + (3-x-3*x^2)*P(n-2,x) + (x-2*x^2)*d/dx P(n-2,x) + (1+3*x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 4, with P(0,x) = 2, P(1,x) = 5 + 3*x, P(2,x) = 11 + 17*x + 4*x^2, P(3,x) = 23 + 69*x + 39*x^2 + 5*x^3.
%t Mathematica
P[0, x_] := 2;
P[1, x_] := 5 + 3*x;
P[2, x_] := 11 + 17*x + 4*x^2;
P[3, x_] := 23 + 69*x + 39*x^2 + 5*x^3;
P[n_, x_] := P[n, x] = Expand[-(2 - x - x^2)*P[n - 4, x] - (x - x^2)*D[P[n - 4, x], x] - (1 + 3*x - x^3)*P[n - 3, x] - (x + x^2 - x^3)*D[P[n - 3, x], x] + (3 - x - 3*x^2)*P[n - 2, x] + (x - 2*x^2)*D[P[n - 2, x], x] + (1 + 3*x)*P[n - 1, x] + x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 5 + 3*x; v[3] = 11 + 17*x + 4*x^2; v[4] = 23 + 69*x + 39*x^2 + 5*x^3; for(n = 4, N, my(i = n + 1); v[i] = -(2 - x - x^2)*v[i - 4] - (x - x^2)*deriv(v[i - 4], x) - (1 + 3*x - x^3)*v[i - 3] - (x + x^2 - x^3)*deriv(v[i - 3], x) + (3 - x - 3*x^2)*v[i - 2] + (x - 2*x^2)*deriv(v[i - 2], x) + (1 + 3*x)*v[i - 1] + 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 - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-34950064e1dbba0d
  • Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
    All coefficients of each polynomial have the same sign (after x -> -x when alternating).
    derived from the cited steps by a stated theorem: e-34950064e1dbba0d
  • Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
    Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
    derived from the cited steps by a stated theorem: e-34950064e1dbba0d

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

No candidate equalities or related sequences.

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.