Real-rooted atlas

A038327 → Pₙ + Pₙ₊₁

nonnegative coefficients only rational roots

Proved for all n: all roots real what this means

Recurrence
\(P_{n}(x) = \left(12 + x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 13 + x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15Pn(1)
013114
1156251182
218724563712366
322464734490049130758
4269568110592181441488611399854
5323481615966723283203600022207315198102
63881779222394880553651276032062640309685167575326
746581350430755635288833024146603521512000997924116971878479238
85589762048415648972813735526402647572483280435227095041491845280109111420230094
9670771445765546763878420639121408455063961665840947265318400449971221254465881211148462991222
10804925734912732688809984303137095680752467968001245155328014422302721193149447050240291600804013311930018885886
119659108818944959719145472043703339581441206098657280224665436160297583165442874009600203917824105494403880809636145125090245516518
121159093058273281248254062755846204119895244818843517845504390208389120058176523468864246431744532102348833051110415206400503712113761571326173191714734
1313909116699279361613814181134336869319793704960288163413098496656685245399041088326670745613527224156161280987136009287156736512987904212509446402241326016914240251492291542
1416690940039135232207566818435399681204565170559385643272807508869121076185707577344196267725029376271159356948482889906978816239544594432154430115847679992322893363279934415288181155123269399790046

Computed in your browser ?

Technical details
Node id
n-a8fee83659838741 · revision 1 · JSON
Definition source
derived:obreschkoff_combination
Fingerprint
f6-264c50b8691afb5b

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 ?

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:          13,           1;
  1:         156,          25,           1;
  2:        1872,         456,          37,          1;
  3:       22464,        7344,         900,         49,         1;
  4:      269568,      110592,       18144,       1488,        61,        1;
  5:     3234816,     1596672,      328320,      36000,      2220,       73,       1;
  6:    38817792,    22394880,     5536512,     760320,     62640,     3096,      85,      1;
  7:   465813504,   307556352,    88833024,   14660352,   1512000,    99792,    4116,     97,    1;
  8:  5589762048,  4156489728,  1373552640,  264757248,  32804352,  2709504,  149184,   5280,  109,   1;
  9: 67077144576, 55467638784, 20639121408, 4550639616, 658409472, 65318400, 4499712, 212544, 6588, 121, 1;
%F Formula
The row polynomials satisfy P(n,x) = (12+x)*P(n-1,x) for n >= 1, with P(0,x) = 13 + x.
%t Mathematica
P[0, x_] := 13 + x;
P[n_, x_] := P[n, x] = Expand[(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] = 13 + x; for(n = 1, N, my(i = n + 1); v[i] = (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-ea24c5115718581a

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.