Real-rooted atlas

A133087: A133080 * A007318.

A133087 nonnegative coefficients only rational roots

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

Recurrence from OEIS A133087
\(P_{n}(x) = \left(1 + 2\,x + x^{2}\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1213
21214
3254112
41464116
52916146148
6161520156164
72133655502781192
818285670562881256
92176414019618211244101768
1011045120210252210120451011024
11221100285540714672450210651213072
1211266220495792924792495220661214096
13225144506121020792640250817829353529014112288
141149136410012002300334323003200210013649114116384

Generated b-files: A084221 (row sums), A133087 ?

Computed in your browser ?

Other recurrences

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

\(P_{n}(x) = \left(5 + 3\,x - 3\,n - 2\,n x\right)\,P_{n-1}(x) + \left(3 + 5\,x + 2\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
Technical details
Node id
n-d75669a8cd7f9393 · revision 2 · JSON
Definition source
oeis_catalog:A133087
derived:shift_x_plus_one (equal by e-13f5d778c9f89527)
Fingerprint
f6-298f80c143fc4de2

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): 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: 2,  1;
  2: 1,  2,  1;
  3: 2,  5,  4,   1;
  4: 1,  4,  6,   4,   1;
  5: 2,  9, 16,  14,   6,   1;
  6: 1,  6, 15,  20,  15,   6,   1;
  7: 2, 13, 36,  55,  50,  27,   8,  1;
  8: 1,  8, 28,  56,  70,  56,  28,  8,  1;
  9: 2, 17, 64, 140, 196, 182, 112, 44, 10, 1;
%F Formula
The row polynomials satisfy P(n,x) = (1+2*x+x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[(1 + 2*x + x^2)*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = (1 + 2*x + x^2)*v[i - 2]); 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.

A133087: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0: 1;
  1: 2,  1;
  2: 1,  2,  1;
  3: 2,  5,  4,   1;
  4: 1,  4,  6,   4,   1;
  5: 2,  9, 16,  14,   6,   1;
  6: 1,  6, 15,  20,  15,   6,   1;
  7: 2, 13, 36,  55,  50,  27,   8,  1;
  8: 1,  8, 28,  56,  70,  56,  28,  8,  1;
  9: 2, 17, 64, 140, 196, 182, 112, 44, 10, 1;
%F Formula
The row polynomials satisfy P(n,x) = (1+2*x+x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[(1 + 2*x + x^2)*P[n - 2, x]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = (1 + 2*x + x^2)*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); 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-3537980ff71b1a89, e-aa3a6b002de14e0c
  • 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-28a1cf3333f63fa4, e-3537980ff71b1a89, e-aa3a6b002de14e0c
  • 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-28a1cf3333f63fa4, e-3537980ff71b1a89, e-aa3a6b002de14e0c
  • 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-28a1cf3333f63fa4

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 14 proof steps

Same fingerprint (1)

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