Real-rooted atlas

A135838 → basis A013609

palindromic γ-positive nonnegative coefficients only rational roots

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

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1448
2816832
332969632256
464256384256641024
525612802560256012802568192
651230727680102407680307251232768
720481433643008716807168043008143362048262144
84096327681146882293762867202293761146883276840961048576
9163841474565898241376256206438420643841376256589824147456163848388608
103276832768014745603932160688128082575366881280393216014745603276803276833554432
111310721441792720896021626880432537606055526460555264432537602162688072089601441792131072268435456
1226214431457281730150457671680129761280207618048242221056207618048129761280576716801730150431457282621441073741824
13104857613631488817889282998927367497318401349517312179935641617993564161349517312749731840299892736817889281363148810485768589934592
14209715229360128190840832763363328209924915241984983046297747456719742566462977474564198498304209924915276336332819084083229360128209715234359738368

Computed in your browser ?

Technical details
Node id
n-e5e748a2234b1a03 · revision 1 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-77a7ed3b1d5f257e

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: 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 ?

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,      4;
  2:     8,     16,      8;
  3:    32,     96,     96,      32;
  4:    64,    256,    384,     256,      64;
  5:   256,   1280,   2560,    2560,    1280,     256;
  6:   512,   3072,   7680,   10240,    7680,    3072,     512;
  7:  2048,  14336,  43008,   71680,   71680,   43008,   14336,   2048;
  8:  4096,  32768, 114688,  229376,  286720,  229376,  114688,  32768,   4096;
  9: 16384, 147456, 589824, 1376256, 2064384, 2064384, 1376256, 589824, 147456, 16384;
%F Formula
The row polynomials satisfy P(n,x) = (8+16*x+8*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 4*x;
P[n_, x_] := P[n, x] = Expand[(8 + 16*x + 8*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 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (8 + 16*x + 8*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-56a3ed705651b472
  • 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-56a3ed705651b472
  • 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-56a3ed705651b472

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 (59)

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