Real-rooted atlas

A038333 → Pₙ + Pₙ₊₁

nonnegative coefficients only rational roots

Proved for all n: all roots real what this means

Recurrence
\(P_{n}(x) = \left(12 + 7\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1872 + 3192\,x + 1813\,x^{2} + 343\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10Pn(1)
01872319218133437220
1224645140844100168072401137180
2269568774144889056510384146461168072606420
332348161117670416087680123480005330220122691111764949521980
4388177921567641602712890882607897601503986405203447210000165823543940917620
54658135042152894464435281817650285007363630312000167720414448424328479883671576480117877434780
65589762048290954280966730407936090811736064787632491524553863372817551348416434830704062836330940353607339671260820
76707714457638827347148810113169489921560869388288158084114227210978063488005293866170881750391233923797850898848827864472824752496453753955580

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-bb1f3df8492d87c4 · revision 1 · JSON
Definition source
derived:obreschkoff_combination
Fingerprint
f6-ead18c54fb42257e

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 ?

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:          1872,           3192,            1813,             343;
  1:         22464,          51408,           44100,           16807,            2401;
  2:        269568,         774144,          889056,          510384,          146461,           16807;
  3:       3234816,       11176704,        16087680,        12348000,         5330220,         1226911,          117649;
  4:      38817792,      156764160,       271289088,       260789760,       150398640,        52034472,        10000165,          823543;
  5:     465813504,     2152894464,      4352818176,      5028500736,      3630312000,      1677204144,       484243284,        79883671,        5764801;
  6:    5589762048,    29095428096,     67304079360,     90811736064,     78763249152,     45538633728,     17551348416,      4348307040,      628363309,       40353607;
  7:   67077144576,   388273471488,   1011316948992,   1560869388288,   1580841142272,   1097806348800,    529386617088,    175039123392,    37978508988,     4882786447,     282475249;
  8:  804925734912,  5128821669888,  14853717688320,  25809651302400,  29896179425280,  24239564181504,  14037283846656,   5806175800320,  1681015971600,   324443000280,   37569208117,   1977326743;
  9: 9659108818944, 67180340183040, 214146363949056, 413691839447040, 539421712220160, 500148026155008, 338124355430400, 167935096530432, 60815422261440, 15660427804560, 2721931499364, 286712377735, 13841287201;
%F Formula
The row polynomials satisfy P(n,x) = (12+7*x)*P(n-1,x) for n >= 1, with P(0,x) = 1872 + 3192*x + 1813*x^2 + 343*x^3.
%t Mathematica
P[0, x_] := 1872 + 3192*x + 1813*x^2 + 343*x^3;
P[n_, x_] := P[n, x] = Expand[(12 + 7*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] = 1872 + 3192*x + 1813*x^2 + 343*x^3; for(n = 1, N, my(i = n + 1); v[i] = (12 + 7*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-946cc0d28b50f639

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.