Real-rooted atlas

A027467 → Pₙ + Pₙ₊₁

nonnegative coefficients only rational roots

Proved for all n: all roots real what this means

Recurrence
\(P_{n}(x) = \left(15 + x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 16 + x\)

Rows ?

nx0x1x2x3x4x5x6x7x8Pn(1)
016117
1240311272
236007054614352
35400014175139561169632
48100002666253510023107611114112
512150000480937579312569750345091117825792
61822500008429062516706250183937512150048151061285212672
727337500001446609375334884375442968753661875193725640512114563402752

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-131a01f7fbbf225f · revision 1 · JSON
Definition source
derived:obreschkoff_combination
Fingerprint
f6-365d4eae33c635d7

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:           16,            1;
  1:          240,           31,            1;
  2:         3600,          705,           46,           1;
  3:        54000,        14175,         1395,          61,          1;
  4:       810000,       266625,        35100,        2310,         76,         1;
  5:     12150000,      4809375,       793125,       69750,       3450,        91,        1;
  6:    182250000,     84290625,     16706250,     1839375,     121500,      4815,      106,      1;
  7:   2733750000,   1446609375,    334884375,    44296875,    3661875,    193725,     6405,    121,     1;
  8:  41006250000,  24432890625,   6469875000,   999337500,   99225000,   6567750,   289800,   8220,   136,   1;
  9: 615093750000, 407499609375, 121481015625, 21459937500, 2487712500, 197741250, 10914750, 413100, 10260, 151, 1;
%F Formula
The row polynomials satisfy P(n,x) = (15+x)*P(n-1,x) for n >= 1, with P(0,x) = 16 + x.
%t Mathematica
P[0, x_] := 16 + x;
P[n_, x_] := P[n, x] = Expand[(15 + 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] = 16 + x; for(n = 1, N, my(i = n + 1); v[i] = (15 + 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-e331e686933fdde7

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.

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.