Real-rooted atlas

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 ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17Pn(1)
025548413311872
1277108966551464122464
229919361863487846161051269568
32121302539930292820112735717715613234816
421434356732057320504348377141724881948717138817792
52165592912112115373051240092762004635175384539214358881465813504
621877744186340286963629311282198414832857435524214358881023579476915589762048
722099801271524491937660877278520838934303999867611575379574259374246012593742460167077144576
822311210037933579061401149904141190488992876922695045015365010153266599915311249095212285311670611804925734912
9225314641512435120788252019579542455383546218646058621414768614606484356150251997181694277370905171794331384283767219659108818944
102275174246734861771561033482502946769210404887382486838198752594222046810910859129973459552256780503549904393799727409434522712143931115909305827328
112297204498651502512395652969673983599963591003199563089195995994906406543876447333814654614871261077584100623263965511789845178406821589653797498335832411390911669927936
1223192371610900893464060680606025514186660488192279916257202311911887816522139470837103853448102404493303878486419171358189368966641082027451277896075997337331856417724816941565116690940039135232
132341272251350965466315851187106921230533232933483331816254138198197705387764497784952855969822816111635691807612863713992455704027522957603575714115512662275337375357101321888006606745949729863572161200291280469622784
142363309761650440614922001700054356361114994246019197378487969863195580842966299532507121379298450564777258612710584251325344407817681014967737031571403037998668665928006349417217511789528270951375442988985054470284992937712403495365635473408

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 ?

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

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.