A038326 → Pₙ + Pₙ₊₁

nonnegative coefficients only rational roots

Proved for all n: all roots real what this means

Recurrence

\(P_{n}(x) = \left(11 + 12\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 12 + 12\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	P_n(1)
0	12	12								24
1	132	276	144							552
2	1452	4620	4896	1728						12696
3	15972	68244	109296	77760	20736					292008
4	175692	942348	2021184	2166912	1161216	248832				6716184
5	1932612	12474132	33541200	48090240	38776320	16671744	2985984			154472232
6	21258732	160406796	518642784	931487040	1003622400	648705024	232906752	35831808		3552861336
7	233846052	2019579540	7629952176	16470070848	22217690880	19179224064	10346434560	3189030912	429981696	81715810728

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-00954e931f2325b9 · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-c0bbc5ef1b77e180

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 ?

A168797: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A168749: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A168845: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A168701: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A167938: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A168941: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A168893: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A168989: 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:          12,           12;
  1:         132,          276,           144;
  2:        1452,         4620,          4896,          1728;
  3:       15972,        68244,        109296,         77760,         20736;
  4:      175692,       942348,       2021184,       2166912,       1161216,         248832;
  5:     1932612,     12474132,      33541200,      48090240,      38776320,       16671744,       2985984;
  6:    21258732,    160406796,     518642784,     931487040,    1003622400,      648705024,     232906752,      35831808;
  7:   233846052,   2019579540,    7629952176,   16470070848,   22217690880,    19179224064,   10346434560,    3189030912,     429981696;
  8:  2572306572,  25021527564,  108164428416,  272730205440,  442035449856,   477583755264,  343961468928,  159236554752,   42998169600,   5159780352;
  9: 28295372292, 306104482068, 1490067043344, 4298005400832, 8135152413696, 10557846706176, 9514581221376, 5879139729408, 2383818522624, 572735619072, 61917364224;

%F Formula

The row polynomials satisfy P(n,x) = (11+12*x)*P(n-1,x) for n >= 1, with P(0,x) = 12 + 12*x.

%t Mathematica

P[0, x_] := 12 + 12*x;
P[n_, x_] := P[n, x] = Expand[(11 + 12*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] = 12 + 12*x; for(n = 1, N, my(i = n + 1); v[i] = (11 + 12*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-4797824a301b09a2

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-4797824a301b09a2

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*12^j.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check liu_wang e-da30941d57c5aa28 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - parent_provers periodic_product e-f0c12876ab8620d1 root: theorem check
    
    Parent-row provers of polytool polytool ?
- premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*12^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.