Real-rooted atlas

A120909 → Pₙ + Pₙ₊₁

nonnegative coefficients only rational roots

Proved for all n: all roots real what this means

Recurrence
\(P_{n}(x) = \left(1 + 2\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 6 + 6\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15Pn(1)
06612
16181236
26304824108
364210812048324
465419233628896972
56663007209606721922916
667843213202400259215363848748
76905882184504073926720345676826244
86102768336094081747221504168967680153678732
9611497248961612836288564485990441472168963072236196
10612612006840259206854412902417280016128099840368646144708588
116138145292403960012038426611243084850688042240023654479872122882125764
12615017281214458080199584506880963072136857614361601081344552960172032245766377292
136162202815600823683157449060481976832329472041733123953664271564812779523686404915219131876
146174235219656113568480480153753637889287248384107627521230028810622976670924829245447864329830457395628

Computed in your browser ?

Technical details
Node id
n-4a90276c6d0931df · revision 1 · JSON
Definition source
derived:obreschkoff_combination
Fingerprint
f6-04668acf2e74f66d

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (affine: slope 1, intercept 1)
  • 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: 6,   6;
  1: 6,  18,  12;
  2: 6,  30,  48,   24;
  3: 6,  42, 108,  120,    48;
  4: 6,  54, 192,  336,   288,    96;
  5: 6,  66, 300,  720,   960,   672,   192;
  6: 6,  78, 432, 1320,  2400,  2592,  1536,   384;
  7: 6,  90, 588, 2184,  5040,  7392,  6720,  3456,   768;
  8: 6, 102, 768, 3360,  9408, 17472, 21504, 16896,  7680,  1536;
  9: 6, 114, 972, 4896, 16128, 36288, 56448, 59904, 41472, 16896, 3072;
%F Formula
The row polynomials satisfy P(n,x) = (1+2*x)*P(n-1,x) for n >= 1, with P(0,x) = 6 + 6*x.
%t Mathematica
P[0, x_] := 6 + 6*x;
P[n_, x_] := P[n, x] = Expand[(1 + 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] = 6 + 6*x; for(n = 1, N, my(i = n + 1); v[i] = (1 + 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, +inf)
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-6efef9a3e4d28a62

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 (1)

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

References

  1. [Fis06] Steve Fisk. Polynomials, roots, and interlacing. arXiv:math/0612833. arXiv
  2. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  3. [Obr63] Nikola Obreschkoff. Verteilung und berechnung der nullstellen reeller polynome. Hochschulbücher für mathematik, VEB Deutscher Verlag der Wissenschaften, 1963.