Real-rooted atlas

basis-map coefficients of A112227 → x ↦ x+1

palindromic γ-positive nonnegative coefficients only rational roots

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

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
11214
213318
31464116
41510105132
5161520156164
6172135352171128
718285670562881256
8193684126126843691512

Show all 15 stored rows

Generated b-files: A020707 (row sums) ?

Computed in your browser ?

Technical details
Node id
n-dd649f176537f8e7 · revision 1 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-77a7ed3b1d5f257e

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 1)
  • 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 ?

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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1,  2,  1;
   2: 1,  3,  3,   1;
   3: 1,  4,  6,   4,   1;
   4: 1,  5, 10,  10,   5,   1;
   5: 1,  6, 15,  20,  15,   6,   1;
   6: 1,  7, 21,  35,  35,  21,   7,   1;
   7: 1,  8, 28,  56,  70,  56,  28,   8,   1;
   8: 1,  9, 36,  84, 126, 126,  84,  36,   9,  1;
   9: 1, 10, 45, 120, 210, 252, 210, 120,  45, 10,  1;
  10: 1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1;
%F Formula
The row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) for n >= 2, with P(1,x) = 1 + 2*x + x^2.
%t Mathematica
P[1, x_] := 1 + 2*x + x^2;
P[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1 + 2*x + x^2; for(n = 2, N, my(i = n); 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, v[i])))};
T(10)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 1, 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-cb9b2e4b7b317e85
  • Coefficients of one sign. Every nonzero row, n >= 1: 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-cb9b2e4b7b317e85, e-d82776c4e84d5954
  • Hadamard kernel (Malo). Rows P_n, n >= 1, 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-cb9b2e4b7b317e85, e-d82776c4e84d5954
  • Real-rooted. Every row P_n, n >= 1, 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-d82776c4e84d5954

Proof chains

2 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

Same fingerprint (59)

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