A038226 → Pₙ + Pₙ₊₁

nonnegative coefficients only rational roots

Proved for all n: all roots real what this means

Recurrence

\(P_{n}(x) = \left(3 + 8\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 36 + 264\,x + 640\,x^{2} + 512\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	P_n(1)
0	36	264	640	512								1452
1	108	1080	4032	6656	4096							15972
2	324	4104	20736	52224	65536	32768						175692
3	972	14904	95040	322560	614400	622592	262144					1932612
4	2916	52488	404352	1728000	4423680	6782976	5767168	2097152				21258732
5	8748	180792	1632960	8418816	27095040	55738368	71565312	52428800	16777216			233846052
6	26244	612360	6345216	38320128	148635648	383975424	660602880	729808896	469762048	134217728		2572306572
7	78732	2047032	23934528	165722112	752467968	2341011456	5053612032	7474249728	7247757312	4160749568	1073741824	28295372292

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-18347f928d4972a4 · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-07069fdd2e8aedf5

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 ?

A168833: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A168977: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A169025: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A003954: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A168881: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A168785: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A168929: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A168737: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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:     36,      264,       640,        512;
  1:    108,     1080,      4032,       6656,        4096;
  2:    324,     4104,     20736,      52224,       65536,       32768;
  3:    972,    14904,     95040,     322560,      614400,      622592,       262144;
  4:   2916,    52488,    404352,    1728000,     4423680,     6782976,      5767168,      2097152;
  5:   8748,   180792,   1632960,    8418816,    27095040,    55738368,     71565312,     52428800,     16777216;
  6:  26244,   612360,   6345216,   38320128,   148635648,   383975424,    660602880,    729808896,    469762048,    134217728;
  7:  78732,  2047032,  23934528,  165722112,   752467968,  2341011456,   5053612032,   7474249728,   7247757312,   4160749568,   1073741824;
  8: 236196,  6770952,  88179840,  688642560,  3583180800, 13042778112,  33888927744,  62851645440,  81537269760,  70464307200,  36507222016,   8589934592;
  9: 708588, 22202424, 318707136, 2771366400, 16258682880, 67793780736, 206009008128, 459666358272, 747424972800, 863691079680, 673236123648, 317827579904, 68719476736;

%F Formula

The row polynomials satisfy P(n,x) = (3+8*x)*P(n-1,x) for n >= 1, with P(0,x) = 36 + 264*x + 640*x^2 + 512*x^3.

%t Mathematica

P[0, x_] := 36 + 264*x + 640*x^2 + 512*x^3;
P[n_, x_] := P[n, x] = Expand[(3 + 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] = 36 + 264*x + 640*x^2 + 512*x^3; for(n = 1, N, my(i = n + 1); v[i] = (3 + 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-c18c7d7bde35dd59

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

obreschkoff_combination e-c18c7d7bde35dd59

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^j.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check liu_wang e-fdf6b848b46b3343 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - parent_provers periodic_product e-711ca5eff5ba56ea root: theorem check
    
    Parent-row provers of polytool polytool ?
- premise (sequence): Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^j.
  uses: Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
  
  no recorded property covers this requirement

Related sequences ?

No candidate equalities or related sequences.

References

[Fis06] Steve Fisk. Polynomials, roots, and interlacing. arXiv:math/0612833. arXiv
[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
[Obr63] Nikola Obreschkoff. Verteilung und berechnung der nullstellen reeller polynome. Hochschulbücher für mathematik, VEB Deutscher Verlag der Wissenschaften, 1963.