Real-rooted atlas

basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle. shifted by x -> -x + 0

palindromic γ-positive nonnegative coefficients only rational roots

Proved for all n: all roots real, in [0, ∞) · all roots real, in [−1, 0] · coefficients of one sign · alternating coefficient signs · Hadamard kernel what this means

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
10011
2000011
300000011
40000000011
5000000000011
600000000000011
70000000000000011

Show all 15 stored rows

Computed in your browser ?

Other recurrences

X ↦ x/(x+1) substitution · proved equal for all n (proof step)

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(x^{2} + 2\,x^{3} + x^{4}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
Technical details
Node id
n-cdb0db4ad473ee28 · revision 2 · JSON
Definition source
derived:root_bound_shift:derived:basis_map_coefficients_of:oeis_catalog:A297191
derived:mobius_x_over_x_plus_one (equal by e-730ec77fe53e082b)
Fingerprint
f6-edf6954a4cd8adfc

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, intercept 0)
  • Palindromic rows: yes
  • γ-positive (gamma_positive): yes
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): yes

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: 1;
  1: 0, 0, 1;
  2: 0, 0, 0, 0, 1;
  3: 0, 0, 0, 0, 0, 0, 1;
  4: 0, 0, 0, 0, 0, 0, 0, 0, 1;
  5: 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  6: 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  7: 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  8: 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  9: 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
%F Formula
The row polynomials satisfy P(n,x) = x^2*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[x^2*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] = x^2*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-1991a148b9538b93
  • 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-12303ad0e0b9f8eb, e-1991a148b9538b93
  • 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-12303ad0e0b9f8eb, e-1991a148b9538b93
  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in [0, +inf)
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-0ca777a455cc19d8
  • Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n(-x) 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-0ca777a455cc19d8
  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-12303ad0e0b9f8eb

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 18 proof steps

Same fingerprint (75)

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