Real-rooted atlas

A038334 → Pₙ + Pₙ₊₁

nonnegative coefficients only rational roots

Proved for all n: all roots real what this means

Recurrence
\(P_{n}(x) = \left(12 + 8\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1872 + 3648\,x + 2368\,x^{2} + 512\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10Pn(1)
01872364823685128400
1224645875257600250884096168000
22695688847361161216761856249856327683360000
332348161277337621012480184320009093120239206426214467200000
4388177921791590403543367683892838402565734401014497282228224020971521344000000
54658135042460450816568531353675061002246193152000326998425610789847042034237441677721626880000000
6558976204833251917824879073689601355557109761343666257928878502707239107690496110729625601828716544134217728537600000000
7670771445764437411102721320903770112232992748339226968451973122140353331200117957250252844573707468811052829900816240345088107374182410752000000000

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-ac84be3513d8b44f · revision 1 · JSON
Definition source
derived:obreschkoff_combination
Fingerprint
f6-e26fe1444ee45b76

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,           3648,            2368,             512;
  1:         22464,          58752,           57600,           25088,            4096;
  2:        269568,         884736,         1161216,          761856,          249856,           32768;
  3:       3234816,       12773376,        21012480,        18432000,         9093120,         2392064,          262144;
  4:      38817792,      179159040,       354336768,       389283840,       256573440,       101449728,        22282240,         2097152;
  5:     465813504,     2460450816,      5685313536,      7506100224,      6193152000,      3269984256,      1078984704,       203423744,        16777216;
  6:    5589762048,    33251917824,     87907368960,    135555710976,    134366625792,     88785027072,     39107690496,     11072962560,      1828716544,      134217728;
  7:   67077144576,   443741110272,   1320903770112,   2329927483392,   2696845197312,   2140353331200,   1179572502528,    445737074688,    110528299008,    16240345088,     1073741824;
  8:  804925734912,  5861510479872,  19400774123520,  38526359961600,  51001562234880,  47259001552896,  31277696679936,  14785424916480,   4892236185600,  1079110533120,   142807662592,    8589934592;
  9: 9659108818944, 76777531637760, 279701373321216, 617522512527360, 920229626511360, 975120516513792, 753404372582400, 427646672437248, 176990233559040, 52087215882240, 10346576216064, 1245540515840, 68719476736;
%F Formula
The row polynomials satisfy P(n,x) = (12+8*x)*P(n-1,x) for n >= 1, with P(0,x) = 1872 + 3648*x + 2368*x^2 + 512*x^3.
%t Mathematica
P[0, x_] := 1872 + 3648*x + 2368*x^2 + 512*x^3;
P[n_, x_] := P[n, x] = Expand[(12 + 8*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 + 3648*x + 2368*x^2 + 512*x^3; for(n = 1, N, my(i = n + 1); v[i] = (12 + 8*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-acb1eb270a760ac4

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.