Real-rooted atlas

basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0

palindromic only rational roots

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

Recurrence
\(P_{n}(x) = \left(-1 - x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1-1-1-2
21214
3-1-3-3-1-8
41464116
5-1-5-10-10-5-1-32
6161520156164
7-1-7-21-35-35-21-7-1-128
818285670562881256
9-1-9-36-84-126-126-84-36-9-1-512
1011045120210252210120451011024
11-1-11-55-165-330-462-462-330-165-55-11-1-2048
1211266220495792924792495220661214096
13-1-13-78-286-715-1287-1716-1716-1287-715-286-78-13-1-8192
141149136410012002300334323003200210013649114116384

Generated b-files: A122803 (row sums) ?

Computed in your browser ?

Technical details
Node id
n-74c1e94669ff21ff · revision 2 · JSON
Definition source
derived:root_bound_shift:derived:basis_map_coefficients_of:oeis_catalog:A113214
Fingerprint
f6-77a7ed3b1d5f257e

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: yes
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): no
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): yes

OEIS matches ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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:  1;
  1: -1, -1;
  2:  1,  2,   1;
  3: -1, -3,  -3,  -1;
  4:  1,  4,   6,   4,    1;
  5: -1, -5, -10, -10,   -5,   -1;
  6:  1,  6,  15,  20,   15,    6,   1;
  7: -1, -7, -21, -35,  -35,  -21,  -7,  -1;
  8:  1,  8,  28,  56,   70,   56,  28,   8,  1;
  9: -1, -9, -36, -84, -126, -126, -84, -36, -9, -1;
%F Formula
The row polynomials satisfy P(n,x) = -(1+x)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(1 + 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] = 1; for(n = 1, N, my(i = n + 1); v[i] = -(1 + 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)

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-60aa79917842b999, e-c62e3171974f4f9c, e-c76ca1e6cd597d3b
  • 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-60aa79917842b999, e-c58069640929a62b, e-c62e3171974f4f9c, e-c76ca1e6cd597d3b
  • 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-60aa79917842b999, e-c58069640929a62b, e-c62e3171974f4f9c, e-c76ca1e6cd597d3b
  • 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-c58069640929a62b

Proof chains

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

Used in 16 proof steps

Same fingerprint (59)

Equal up to content, sign, reversal and a small index shift.