Real-rooted atlas

A008277 → θ+2 (shift +1) → x ↦ x+1

nonnegative coefficients

Proved for all n: all roots real, in (−∞, −1] · coefficients of one sign · Hadamard kernel what this means

Recurrence
\(P_{n}(x) = \left(1 - x - x^{2}\right)\,P_{n-2}(x) + \left(-x - x^{2}\right)\,P_{n-2}'(x) + \left(2 + 2\,x\right)\,P_{n-1}(x) + \left(1 + x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 2\)
\(P_{1}(x) = 5 + 3\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
022
1538
21517436
35284395180
4203419286746988
58772183194575512575892
64140119901308467401695195837852
7211476954189348571201928533952879260132

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-ebbce5356dede62c · revision 1 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-1b51648b831a15d9

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:     15,      17,       4;
  3:     52,      84,      39,       5;
  4:    203,     419,     286,      74,       6;
  5:    877,    2183,    1945,     755,     125,      7;
  6:   4140,   11990,   13084,    6740,    1695,    195,      8;
  7:  21147,   69541,   89348,   57120,   19285,   3395,    287,     9;
  8: 115975,  425473,  626630,  476714,  201796,  48062,   6244,   404,  10;
  9: 678570, 2740482, 4538031, 3992189, 2028831, 610365, 107814, 10746, 549, 11;
%F Formula
The row polynomials satisfy P(n,x) = (1-x-x^2)*P(n-2,x) - (x+x^2)*d/dx P(n-2,x) + (2+2*x)*P(n-1,x) + (1+x)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 2, P(1,x) = 5 + 3*x.
%t Mathematica
P[0, x_] := 2;
P[1, x_] := 5 + 3*x;
P[n_, x_] := P[n, x] = Expand[(1 - x - x^2)*P[n - 2, x] - (x + x^2)*D[P[n - 2, x], x] + (2 + 2*x)*P[n - 1, x] + (1 + 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; for(n = 2, N, my(i = n + 1); v[i] = (1 - x - x^2)*v[i - 2] - (x + x^2)*deriv(v[i - 2], x) + (2 + 2*x)*v[i - 1] + (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, -1]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-0a406e2d3e1f653c
  • 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-0a406e2d3e1f653c
  • 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-0a406e2d3e1f653c

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.