Real-rooted atlas

A038319 → Pₙ + Pₙ₊₁

nonnegative coefficients only rational roots

Proved for all n: all roots real what this means

Recurrence
\(P_{n}(x) = \left(11 + 5\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1452 + 1925\,x + 850\,x^{2} + 125\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10Pn(1)
0145219258501254352
1159722843518975562562569632
21756923926453509001567503500031251114112
3193261251975555823125347875011687502093751562517825792
421258732668361659004215067381875302500008146875121875078125285212672
5233846052841491475132464447511914113756696593752408656255414062569531253906254563402752
62572306572104256364851877854660019728747500133233100005997818750179987500034718750039062500195312573014444032
7282953722921275435341952586921950253109089555002452001475001325925562504978771875012818437500216562500021679687597656251168231104512

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-2323ae2e7daf2827 · revision 1 · JSON
Definition source
derived:obreschkoff_combination
Fingerprint
f6-fbb9b0153d17b34b

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:          1452,           1925,            850,            125;
  1:         15972,          28435,          18975,           5625,            625;
  2:        175692,         392645,         350900,         156750,          35000,           3125;
  3:       1932612,        5197555,        5823125,        3478750,        1168750,         209375,          15625;
  4:      21258732,       66836165,       90042150,       67381875,       30250000,        8146875,        1218750,          78125;
  5:     233846052,      841491475,     1324644475,     1191411375,      669659375,      240865625,       54140625,        6953125,        390625;
  6:    2572306572,    10425636485,    18778546600,    19728747500,    13323310000,     5997818750,     1799875000,      347187500,      39062500,      1953125;
  7:   28295372292,   127543534195,   258692195025,   310908955500,   245200147500,   132592556250,    49787718750,    12818437500,    2165625000,    216796875,     9765625;
  8:  311249095212,  1544455737605,  3483331816250,  4713459485625,  4251746400000,  2684518856250,  1210627687500,   389941406250,   87914062500,  13212890625,  1191406250,   48828125;
  9: 3423740047332, 18545258589715, 46038928666775, 69264713423125, 70336507828125, 50788439418750, 26739498843750, 10342493906250, 2916761718750, 584912109375, 79169921875, 6494140625, 244140625;
%F Formula
The row polynomials satisfy P(n,x) = (11+5*x)*P(n-1,x) for n >= 1, with P(0,x) = 1452 + 1925*x + 850*x^2 + 125*x^3.
%t Mathematica
P[0, x_] := 1452 + 1925*x + 850*x^2 + 125*x^3;
P[n_, x_] := P[n, x] = Expand[(11 + 5*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] = 1452 + 1925*x + 850*x^2 + 125*x^3; for(n = 1, N, my(i = n + 1); v[i] = (11 + 5*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-7723a7442c6c45f3

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.