A013613 → Pₙ + Pₙ₊₁

nonnegative coefficients only rational roots

Proved for all n: all roots real what this means

Recurrence

\(P_{n}(x) = \left(1 + 6\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 2 + 6\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	P_n(1)
0	2	6															8
1	2	18	36														56
2	2	30	144	216													392
3	2	42	324	1080	1296												2744
4	2	54	576	3024	7776	7776											19208
5	2	66	900	6480	25920	54432	46656										134456
6	2	78	1296	11880	64800	209952	373248	279936									941192
7	2	90	1764	19656	136080	598752	1632960	2519424	1679616								6588344
8	2	102	2304	30240	254016	1415232	5225472	12317184	16796160	10077696							46118408
9	2	114	2916	44064	435456	2939328	13716864	43670016	90699264	110854656	60466176						322828856
10	2	126	3600	61560	699840	5552064	31352832	125971200	352719360	655050240	725594112	362797056					2259801992
11	2	138	4356	83160	1069200	9751104	64665216	314088192	1108546560	2771366400	4655895552	4716361728	2176782336				15818613944
12	2	150	5184	109296	1568160	16166304	123171840	702079488	2993075712	9422645760	21284093952	32651735040	30474952704	13060694016			110730297608
13	2	162	6084	140400	2223936	25575264	220169664	1441110528	7205552640	27381100032	77819968512	160356298752	226385362944	195910410240	78364164096		775112083256
14	2	174	7056	176904	3066336	38918880	373621248	2762128512	15852215808	70614415872	242106568704	627276109824	1188523155456	1554222587904	1253826625536	470184984576	5425784582792

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Technical details

Node id: n-7d37bf0e0a7dbf4f · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-97b5915faaad55c8

Observed on the stored rows ?

Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (affine: slope 1, intercept 1)
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 ?

A167899: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A168733: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A168781: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A168685: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A168877: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A168829: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A168925: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A003950: 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: 2,   6;
  1: 2,  18,   36;
  2: 2,  30,  144,   216;
  3: 2,  42,  324,  1080,   1296;
  4: 2,  54,  576,  3024,   7776,    7776;
  5: 2,  66,  900,  6480,  25920,   54432,    46656;
  6: 2,  78, 1296, 11880,  64800,  209952,   373248,   279936;
  7: 2,  90, 1764, 19656, 136080,  598752,  1632960,  2519424,  1679616;
  8: 2, 102, 2304, 30240, 254016, 1415232,  5225472, 12317184, 16796160,  10077696;
  9: 2, 114, 2916, 44064, 435456, 2939328, 13716864, 43670016, 90699264, 110854656, 60466176;

%F Formula

The row polynomials satisfy P(n,x) = (1+6*x)*P(n-1,x) for n >= 1, with P(0,x) = 2 + 6*x.

%t Mathematica

P[0, x_] := 2 + 6*x;
P[n_, x_] := P[n, x] = Expand[(1 + 6*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] = 2 + 6*x; for(n = 1, N, my(i = n + 1); v[i] = (1 + 6*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-84f0fe7fd2f0b117

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-84f0fe7fd2f0b117

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): Triangle of coefficients in expansion of (1+6x)^n.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check liu_wang e-1784cac1b85d5f84 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - parent_provers periodic_product e-5658c7be1cbffd3f root: theorem check
    
    Parent-row provers of polytool polytool ?
- premise (sequence): Triangle of coefficients in expansion of (1+6x)^n.
  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.