Real-rooted atlas

A159854 → x ↦ x+1

nonnegative coefficients only rational roots

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

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
01214
1254112
2412136136
3828382581108
416641048841101324
53214427228017061121972
664320688832620292851412916
712870416962352207212044621131618748

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-21d2f90cb223c8f5 · revision 1 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-a3dcade816dfaa44

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 2)
  • 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 ?

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,    2,    1;
  1:   2,    5,    4,     1;
  2:   4,   12,   13,     6,     1;
  3:   8,   28,   38,    25,     8,     1;
  4:  16,   64,  104,    88,    41,    10,    1;
  5:  32,  144,  272,   280,   170,    61,   12,    1;
  6:  64,  320,  688,   832,   620,   292,   85,   14,   1;
  7: 128,  704, 1696,  2352,  2072,  1204,  462,  113,  16,   1;
  8: 256, 1536, 4096,  6400,  6496,  4480, 2128,  688, 145,  18,  1;
  9: 512, 3328, 9728, 16896, 19392, 15456, 8736, 3504, 978, 181, 20, 1;
%F Formula
The row polynomials satisfy P(n,x) = (2+x)*P(n-1,x) for n >= 1, with P(0,x) = 1 + 2*x + x^2.
%t Mathematica
P[0, x_] := 1 + 2*x + x^2;
P[n_, x_] := P[n, x] = Expand[(2 + 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 + 2*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = (2 + 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, -1]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-34caf7309ed0b453
  • 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-34caf7309ed0b453, e-7a44d9ed78450a4e
  • 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-34caf7309ed0b453, e-7a44d9ed78450a4e
  • 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-7a44d9ed78450a4e

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

No candidate equalities or related sequences.