Real-rooted atlas

A038329 → Pₙ + Pₙ₊₁

nonnegative coefficients only rational roots

Proved for all n: all roots real what this means

Recurrence
\(P_{n}(x) = \left(12 + 3\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1872 + 1368\,x + 333\,x^{2} + 27\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10Pn(1)
018721368333273600
12246422032810013238154000
2269568331776163296401764941243810000
33234816479001629548809720001798201773972912150000
4388177926718464049828608205286405073840752328619652187182250000
546581350492266905679949721639582950412247200024249456300056421213965612733750000
65589762048124694691841236197376071484456962657152512658409472108755136115473607151491968341006250000
7670771445761664029163521857520926721228672696325333116723215872371200328029004846483372843223868238164359049615093750000

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-2368212452cf1434 · revision 1 · JSON
Definition source
derived:obreschkoff_combination
Fingerprint
f6-73bcdb549bd5a284

Observed on the stored rows ?

  • Degrees: 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 (affine: slope 1, intercept 3)
  • 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:          1872,           1368,            333,             27;
  1:         22464,          22032,           8100,           1323,             81;
  2:        269568,         331776,         163296,          40176,           4941,           243;
  3:       3234816,        4790016,        2954880,         972000,         179820,         17739,           729;
  4:      38817792,       67184640,       49828608,       20528640,        5073840,        752328,         61965,         2187;
  5:     465813504,      922669056,      799497216,      395829504,      122472000,      24249456,       3000564,       212139,        6561;
  6:    5589762048,    12469469184,    12361973760,     7148445696,     2657152512,     658409472,     108755136,     11547360,      715149,      19683;
  7:   67077144576,   166402916352,   185752092672,   122867269632,    53331167232,   15872371200,    3280290048,    464833728,    43223868,    2381643,     59049;
  8:  804925734912,  2198066429952,  2728233861120,  2031663513600,  1008575815680,  350461956096,   86980594176,  15418874880,  1913187600,  158251320,   7853517,   177147;
  9: 9659108818944, 28791574364160, 39333005623296, 32564663746560, 18197900328960, 7231270920192, 2095152998400, 445968281088, 69214875840, 7638578640, 568996164, 25686315, 531441;
%F Formula
The row polynomials satisfy P(n,x) = (12+3*x)*P(n-1,x) for n >= 1, with P(0,x) = 1872 + 1368*x + 333*x^2 + 27*x^3.
%t Mathematica
P[0, x_] := 1872 + 1368*x + 333*x^2 + 27*x^3;
P[n_, x_] := P[n, x] = Expand[(12 + 3*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] = 1872 + 1368*x + 333*x^2 + 27*x^3; for(n = 1, N, my(i = n + 1); v[i] = (12 + 3*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-bebf949c785e986b

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.