Real-rooted atlas

A038271 → Pₙ + Pₙ₊₁

nonnegative coefficients only rational roots

Proved for all n: all roots real what this means

Recurrence
\(P_{n}(x) = \left(7 + 5\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 8 + 5\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15Pn(1)
08513
1567525156
23928055501251872
3274475957875362562522464
419208668859310064750225003125269568
5134456564235986125918750481250134375156253234816
69411924621925972405011361875796250033468757812507812538817792
76588344370594359117797512815337511254687563240625222031254453125390625465813504
8461184082923577658235430001352963500142859500010054187504716250001421875002500000019531255589762048
93228288562277096395722658982513588459500167649825001418090625083284687503353437500885937500138671875976562567077144576
10225980199217553819045619716107501312521656251852971750001830912562501292038125006511640625022968750000540039062576171875048828125804925734912
1115818613944134175743275521570370475122862321312519533410531252208124668750181988296875011018339062504863632812501526464843753233398437541503906252441406259659108818944
1211073029760810183232726454321871309700112082143442501981650343750025223577946875237798041250001681225218750089137125000003500341796875989570312500190722656250224609375001220703125115909305827328
1377511208325676819143965553534471553112510006685695825019475659578375027564756281562529257651860937523658478593750014645724843750069070955078125244287011718756282910156250111083984375012084960937561035156251390911669927936
145425784582792576489611921652858225807006508771915763633751863630455277500290331591862812534262734443437503118976094609375220812466875000012157829277343755163556835937501661238769531253919042968750064001464843756469726562503051757812516690940039135232

Computed in your browser ?

Technical details
Node id
n-29f5117c899fc989 · revision 1 · JSON
Definition source
derived:obreschkoff_combination
Fingerprint
f6-327bcf57e38a9664

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:         8,          5;
  1:        56,         75,         25;
  2:       392,        805,        550,         125;
  3:      2744,       7595,       7875,        3625,         625;
  4:     19208,      66885,      93100,       64750,       22500,        3125;
  5:    134456,     564235,     986125,      918750,      481250,      134375,      15625;
  6:    941192,    4621925,    9724050,    11361875,     7962500,     3346875,     781250,      78125;
  7:   6588344,   37059435,   91177975,   128153375,   112546875,    63240625,   22203125,    4453125,    390625;
  8:  46118408,  292357765,  823543000,  1352963500,  1428595000,  1005418750,  471625000,  142187500,  25000000,   1953125;
  9: 322828856, 2277096395, 7226589825, 13588459500, 16764982500, 14180906250, 8328468750, 3353437500, 885937500, 138671875, 9765625;
%F Formula
The row polynomials satisfy P(n,x) = (7+5*x)*P(n-1,x) for n >= 1, with P(0,x) = 8 + 5*x.
%t Mathematica
P[0, x_] := 8 + 5*x;
P[n_, x_] := P[n, x] = Expand[(7 + 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] = 8 + 5*x; for(n = 1, N, my(i = n + 1); v[i] = (7 + 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-eee83bb3f7c38b3e

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.