A013618 → Pₙ + Pₙ₊₁

nonnegative coefficients only rational roots

Proved for all n: all roots real what this means

Recurrence

\(P_{n}(x) = \left(1 + 11\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 2 + 55\,x + 484\,x^{2} + 1331\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	P_n(1)
0	2	55	484	1331								1872
1	2	77	1089	6655	14641							22464
2	2	99	1936	18634	87846	161051						269568
3	2	121	3025	39930	292820	1127357	1771561					3234816
4	2	143	4356	73205	732050	4348377	14172488	19487171				38817792
5	2	165	5929	121121	1537305	12400927	62004635	175384539	214358881			465813504
6	2	187	7744	186340	2869636	29311282	198414832	857435524	2143588810	2357947691		5589762048
7	2	209	9801	271524	4919376	60877278	520838934	3039998676	11575379574	25937424601	25937424601	67077144576

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-8a3a4358b42999f7 · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-4adf7795e7695c39

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 ?

A168690: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A168978: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A168834: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A168930: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A168738: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A169026: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A168882: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A168786: row sums, 14 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,  55,   484,   1331;
  1: 2,  77,  1089,   6655,    14641;
  2: 2,  99,  1936,  18634,    87846,    161051;
  3: 2, 121,  3025,  39930,   292820,   1127357,    1771561;
  4: 2, 143,  4356,  73205,   732050,   4348377,   14172488,    19487171;
  5: 2, 165,  5929, 121121,  1537305,  12400927,   62004635,   175384539,    214358881;
  6: 2, 187,  7744, 186340,  2869636,  29311282,  198414832,   857435524,   2143588810,   2357947691;
  7: 2, 209,  9801, 271524,  4919376,  60877278,  520838934,  3039998676,  11575379574,  25937424601,   25937424601;
  8: 2, 231, 12100, 379335,  7906140, 114990414, 1190488992,  8769226950,  45015365010, 153266599915,  311249095212,  285311670611;
  9: 2, 253, 14641, 512435, 12078825, 201957954, 2455383546, 21864605862, 141476861460, 648435615025, 1997181694277, 3709051717943, 3138428376721;

%F Formula

The row polynomials satisfy P(n,x) = (1+11*x)*P(n-1,x) for n >= 1, with P(0,x) = 2 + 55*x + 484*x^2 + 1331*x^3.

%t Mathematica

P[0, x_] := 2 + 55*x + 484*x^2 + 1331*x^3;
P[n_, x_] := P[n, x] = Expand[(1 + 11*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 + 55*x + 484*x^2 + 1331*x^3; for(n = 1, N, my(i = n + 1); v[i] = (1 + 11*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-f27f365b1aa97cdc

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-f27f365b1aa97cdc

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): Triangle of coefficients in expansion of (1+11x)^n.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check liu_wang e-5d158048399db1f0 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - parent_provers periodic_product e-70681fa17f5604ff root: theorem check
    
    Parent-row provers of polytool polytool ?
- premise (sequence): Triangle of coefficients in expansion of (1+11x)^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.