Real-rooted atlas

A038331 → Pₙ + Pₙ₊₁

nonnegative coefficients only rational roots

Proved for all n: all roots real what this means

Recurrence
\(P_{n}(x) = \left(12 + 5\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1872 + 2280\,x + 925\,x^{2} + 125\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17Pn(1)
0187222809251255202
1224643672022500612562588434
22695685529604536001860003812531251503378
3323481679833608208000450000013875002281251562525557426
43881779211197440013841280095040000391500009675000132812578125434476242
54658135041537781760222082560018325440009450000003118500006431250075781253906257386096114
655897620482078244864034338816000330946560002050272000084672000002331000000412500000425781251953125125563633938
7670771445762773381939205159780352005688299520004115059200002041200000007030800000016605000000257343750023632812597656252134581776946
88049257349123663444049920757842739200094058496000007782220800000450696960000018642960000005508000000001139062500001570312500012988281254882812536287890208082
99659108818944479859572736001092583489536001507623321600001404158976000009299473920000044906400000000159310800000004120875000000757968750000941015625007080078125244140625616894133537394
101159093058273286241270313779201551029973811200235543973068800024388024320000001818016358400000100385049600000041570496000000012910590000000029700000000000491906250000055546875000038330078125122070312510487200270135698
1113909116699279368069070905671680217329948426240003602042663731200041042827837440000340102084608000002113628774400000010007712000000000362779560000000010019295000000002075287500000003126093750000032373046875002062988281256103515625178282404592306866
1216690940039135232103783409217699840301141292639846400540910093860864000672616067235840000613336640716800000423686495232000000225773982720000000935721072000000003016213200000000074999925000000001412775000000000195152343750000186621093750001104736328125305175781253030800878069216722
1320029128046962278413288556108080742404132612557766656000799662758952960000010775943276134400000107231200247808000008150921146368000000482772026880000000022517352000000000008298061200000000002408105700000000005445326250000000094057031250000001199707031250000106567382812500588989257812515258789062551523614927176684274
14240349536563547340816947723732045004800562356287472402432001166225938631884800001692944572612608000001825571566780416000001514266538803200000009868724895744000000051159423744000000000212163494400000000007038757440000000000185749200000000000038513475000000000061425000000000000727734375000000060351562500000031280517578125762939453125875901453762003632658

Computed in your browser ?

Technical details
Node id
n-2255a1a798a79b80 · revision 1 · JSON
Definition source
derived:obreschkoff_combination
Fingerprint
f6-5db386616ed16bd2

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,           2280,             925,             125;
  1:         22464,          36720,           22500,            6125,             625;
  2:        269568,         552960,          453600,          186000,           38125,           3125;
  3:       3234816,        7983360,         8208000,         4500000,         1387500,         228125,          15625;
  4:      38817792,      111974400,       138412800,        95040000,        39150000,        9675000,        1328125,          78125;
  5:     465813504,     1537781760,      2220825600,      1832544000,       945000000,      311850000,       64312500,        7578125,        390625;
  6:    5589762048,    20782448640,     34338816000,     33094656000,     20502720000,     8467200000,     2331000000,      412500000,      42578125,      1953125;
  7:   67077144576,   277338193920,    515978035200,    568829952000,    411505920000,   204120000000,    70308000000,    16605000000,    2573437500,    236328125,     9765625;
  8:  804925734912,  3663444049920,   7578427392000,   9405849600000,   7782220800000,  4506969600000,  1864296000000,   550800000000,  113906250000,  15703125000,  1298828125,   48828125;
  9: 9659108818944, 47985957273600, 109258348953600, 150762332160000, 140415897600000, 92994739200000, 44906400000000, 15931080000000, 4120875000000, 757968750000, 94101562500, 7080078125, 244140625;
%F Formula
The row polynomials satisfy P(n,x) = (12+5*x)*P(n-1,x) for n >= 1, with P(0,x) = 1872 + 2280*x + 925*x^2 + 125*x^3.
%t Mathematica
P[0, x_] := 1872 + 2280*x + 925*x^2 + 125*x^3;
P[n_, x_] := P[n, x] = Expand[(12 + 5*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 + 2280*x + 925*x^2 + 125*x^3; for(n = 1, N, my(i = n + 1); v[i] = (12 + 5*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-d496cfbb04db8f55

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.