Real-rooted atlas

A159854 → basis A013611

nonnegative coefficients only rational roots

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

Recurrence
\(P_{n}(x) = \left(2 + 4\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1 + 8\,x + 16\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0181625
12206464150
2448208384256900
381126081600204810245400
416256166456321049610240409632400
53257643521792043520624644915216384194400
66412801100853248158720299008348160229376655361166400
712828162713615052853043212328961892352185139210485762621446998400

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-84c761192e0db3c0 · revision 1 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-23ef07238c102b65

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,     8,     16;
  1:   2,    20,     64,      64;
  2:   4,    48,    208,     384,     256;
  3:   8,   112,    608,    1600,    2048,     1024;
  4:  16,   256,   1664,    5632,   10496,    10240,     4096;
  5:  32,   576,   4352,   17920,   43520,    62464,    49152,    16384;
  6:  64,  1280,  11008,   53248,  158720,   299008,   348160,   229376,    65536;
  7: 128,  2816,  27136,  150528,  530432,  1232896,  1892352,  1851392,  1048576,   262144;
  8: 256,  6144,  65536,  409600, 1662976,  4587520,  8716288, 11272192,  9502720,  4718592,  1048576;
  9: 512, 13312, 155648, 1081344, 4964352, 15826944, 35782656, 57409536, 64094208, 47448064, 20971520, 4194304;
%F Formula
The row polynomials satisfy P(n,x) = (2+4*x)*P(n-1,x) for n >= 1, with P(0,x) = 1 + 8*x + 16*x^2.
%t Mathematica
P[0, x_] := 1 + 8*x + 16*x^2;
P[n_, x_] := P[n, x] = Expand[(2 + 4*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 + 8*x + 16*x^2; for(n = 1, N, my(i = n + 1); v[i] = (2 + 4*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, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-d3bf3843e93b142d
  • 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-d3bf3843e93b142d
  • 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-d3bf3843e93b142d

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

No candidate equalities or related sequences.