Real-rooted atlas

A038305 → Pₙ + Pₙ₊₁

nonnegative coefficients only rational roots

Proved for all n: all roots real what this means

Recurrence
\(P_{n}(x) = \left(10 + 3\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1100 + 960\,x + 279\,x^{2} + 27\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17Pn(1)
01100960279272366
11100012900567011078130758
211000016200095400280804131243399854
3110000019500001440000567000125550148237295198102
41100000022800000202500009990000295650052488051759218767575326
5110000000261000000270900000160650000595350001411830020922301771476561878479238
6110000000029400000003492000000241920000010773000003197880006327720080481605970511968311420230094
711000000000327000000004374000000034668000000180306000006429780000159213600027031320030114990198798359049148462991222
8110000000000360000000000535500000000477900000000284310000000118389600000352107000007479540000111208950011022480065544391771471930018885886
91100000000000393000000000064350000000006385500000000427680000000020368260000007072758000001804275000003355951500044385165003962187902143478753144125090245516518
10110000000000004260000000000076140000000000831600000000006192450000000033198660000000131832360000003926102400000876877650000145063710000172777374001403004240696187711594323326173191714734
1111000000000000045900000000000088920000000000010600200000000008687250000000005177601000000002314283400000007881073200000020547083700000408127005000060796850400065863254600490520043022479954347829694240251492291542
1211000000000000004920000000000000102690000000000001326780000000000011867310000000000778377600000000038675637000000001482392340000000441903033000000102453951600000183234951900002482538058000246641768100169635967207222283191434890755123269399790046
13110000000000000005250000000000000011745000000000000016348500000000000015847650000000000011343969000000000062026965000000000264266145000000008866207350000000235024861500000049059680670000079795866150000991403185500090956127150058113073350231017402743046721716602502197270598
1411000000000000000055800000000000000013320000000000000001987200000000000000207522000000000000016098264000000000009605887200000000004503470400000000001679419170000000005010110820000000011956713912000000226974908160000033852791700000038837708280000330981454800019744096032073609892911291401639315832528564517774

Computed in your browser ?

Technical details
Node id
n-91e781c52666cb4c · revision 1 · JSON
Definition source
derived:obreschkoff_combination
Fingerprint
f6-bf4fb4625505427f

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:          1100,           960,           279,            27;
  1:         11000,         12900,          5670,          1107,            81;
  2:        110000,        162000,         95400,         28080,          4131,           243;
  3:       1100000,       1950000,       1440000,        567000,        125550,         14823,          729;
  4:      11000000,      22800000,      20250000,       9990000,       2956500,        524880,        51759,         2187;
  5:     110000000,     261000000,     270900000,     160650000,      59535000,      14118300,      2092230,       177147,        6561;
  6:    1100000000,    2940000000,    3492000000,    2419200000,    1077300000,     319788000,     63277200,      8048160,      597051,      19683;
  7:   11000000000,   32700000000,   43740000000,   34668000000,   18030600000,    6429780000,   1592136000,    270313200,    30114990,    1987983,     59049;
  8:  110000000000,  360000000000,  535500000000,  477900000000,  284310000000,  118389600000,  35210700000,   7479540000,  1112089500,  110224800,   6554439,   177147;
  9: 1100000000000, 3930000000000, 6435000000000, 6385500000000, 4276800000000, 2036826000000, 707275800000, 180427500000, 33559515000, 4438516500, 396218790, 21434787, 531441;
%F Formula
The row polynomials satisfy P(n,x) = (10+3*x)*P(n-1,x) for n >= 1, with P(0,x) = 1100 + 960*x + 279*x^2 + 27*x^3.
%t Mathematica
P[0, x_] := 1100 + 960*x + 279*x^2 + 27*x^3;
P[n_, x_] := P[n, x] = Expand[(10 + 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] = 1100 + 960*x + 279*x^2 + 27*x^3; for(n = 1, N, my(i = n + 1); v[i] = (10 + 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-e4e37e9713aaf48a

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.