Real-rooted atlas

A038325 → Pₙ + Pₙ₊₁

nonnegative coefficients only rational roots

Proved for all n: all roots real what this means

Recurrence
\(P_{n}(x) = \left(11 + 11\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 12 + 11\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15Pn(1)
0121123
1132253121506
2145242354114133111132
31597262557918395989514641244904
4175692863819169835616690748198961610515387888
5193261211434621281839253704173027378670107904171771561118533536
621258732147039563435804006717482205708624400419859957138181758194871712607737792
7233846052185128124564112792591268614832115687172655124133279276138458865173435821921435888157370231424
825723065722293640026790888165544210071703380312106530736309105506402204069654712866009879242143588810023579476911262145091328
92829537229228059577522912520702239213310558558164574396057527668333324085185644926772254319737706899611884056362642617321937012593742460127767192009216
1031124909521233978026227311685932599065050188916602935995997104678401383502228217341372608509884929726534225375048243609757860159515161296153164365801322285311670611610878224202752
113423740047332407995688973732228284147471917375306685294351647674897778525261744926618531430317218119124862579788125664662160059847212771070614638476222521027470124030737946452191263313842837672113439320932460544
123766114052065248645639839175528999078200902041056394991604288626237261229387560469163658036022296214088185907580061726609313348628459842525757160922537419342578928510080631946027852273043268774727045193368624782434522712143931295665060514131968
134142725457271725765292928036477372500064033015491481024350974639904048133225997349068046898973628876791199629724289458319136254240289666870811848194807797119207849429060165591473900030790899885071409217109715263423500603011394603453510203823093053797498335832416504631331310903296
144556998002998892679752202114001394731682926447182862038876856508420929608207333466918785613304535419588848435220475158381758059782818389339904739729728020980720440186908216673862547589817371292440583617202419458401726906766633391935205151940196136443927555458808729630384723748180064177248169415651143101889288839872512

Computed in your browser ?

Technical details
Node id
n-7a6c33994a6c1ce3 · revision 1 · JSON
Definition source
derived:obreschkoff_combination
Fingerprint
f6-184084f9fbde5b90

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:          12,           11;
  1:         132,          253,           121;
  2:        1452,         4235,          4114,          1331;
  3:       15972,        62557,         91839,         59895,         14641;
  4:      175692,       863819,       1698356,       1669074,        819896,        161051;
  5:     1932612,     11434621,      28183925,      37041730,      27378670,      10790417,       1771561;
  6:    21258732,    147039563,     435804006,     717482205,     708624400,     419859957,     138181758,      19487171;
  7:   233846052,   1851281245,    6411279259,   12686148321,   15687172655,   12413327927,    6138458865,    1734358219,     214358881;
  8:  2572306572,  22936400267,   90888165544,  210071703380,  312106530736,  309105506402,  204069654712,   86600987924,   21435888100,   2357947691;
  9: 28295372292, 280595775229, 1252070223921, 3310558558164, 5743960575276, 6833332408518, 5644926772254, 3197377068996, 1188405636264, 261732193701, 25937424601;
%F Formula
The row polynomials satisfy P(n,x) = (11+11*x)*P(n-1,x) for n >= 1, with P(0,x) = 12 + 11*x.
%t Mathematica
P[0, x_] := 12 + 11*x;
P[n_, x_] := P[n, x] = Expand[(11 + 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] = 12 + 11*x; for(n = 1, N, my(i = n + 1); v[i] = (11 + 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-d2886dc5bcf83686

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.