Real-rooted atlas

A135838 → basis A013610

nonnegative coefficients only rational roots

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence
\(P_{n}(x) = \left(8 + 24\,x + 18\,x^{2}\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 6\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
14610
28241850
332144216108500
4643848648643242500
52561920576086406480194425000
65124608172803456038880233285832125000
720482150496768241920362880326592163296349921250000

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-545ecce3e4f3660f · revision 1 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-f06e51c139cfae05

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 ?

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:     4,      6;
  2:     8,     24,      18;
  3:    32,    144,     216,     108;
  4:    64,    384,     864,     864,      324;
  5:   256,   1920,    5760,    8640,     6480,     1944;
  6:   512,   4608,   17280,   34560,    38880,    23328,     5832;
  7:  2048,  21504,   96768,  241920,   362880,   326592,   163296,    34992;
  8:  4096,  49152,  258048,  774144,  1451520,  1741824,  1306368,   559872,  104976;
  9: 16384, 221184, 1327104, 4644864, 10450944, 15676416, 15676416, 10077696, 3779136, 629856;
%F Formula
The row polynomials satisfy P(n,x) = (8+24*x+18*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 6*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 6*x;
P[n_, x_] := P[n, x] = Expand[(8 + 24*x + 18*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] = 4 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = (8 + 24*x + 18*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.

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-9d8a38d997eb16b9
  • 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-9d8a38d997eb16b9
  • 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-9d8a38d997eb16b9

Proof chains

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

Same fingerprint (23)

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