Real-rooted atlas

A099093 → basis A013610

nonnegative coefficients

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence
\(P_{n}(x) = \left(3 + 9\,x\right)\,P_{n-2}(x) + \left(3 + 9\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 9\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
13912
2126381156
3453518917292016
41711755664210935656126064
564882624155310278912465959049336960
6245737422234738774927141717613581275314414356288
7931516499712400295119767125249491812804314348907478296956318976
8353167132056246072309569859487862118393763522001657414827203943046721728103168
913389330362853036212117560516564690147815728291642521687545256845435315066352353874204899413065728
1050762712771351143569989949160187408439964711946321288240360135483282551624729099583396151093990713486784401121694026752
1119245605319658866406505449396850462431679154383139161481201345457347345201221637410364391293321946426359149931729243313810596091573285109760
12729656121979350030166165802493525108613820318066754086716901715319137576063162513451707472063256502349663432484913493757969802147490980162328242953648120339749638144
1327663363901960353134989556941227509431027564343418183334698741384108158346193412612247695460946962422929044620438416313945852567617078537312079875101144039063605312541865828329262956416974848
1410487977236799008755964250149959169873303040130883151471967843543356871923338324450198984706611471418614080139045666178054086258789149966340726674535876295455393141914781552139802620558095228767924549613399553999355904

Computed in your browser ?

Technical details
Node id
n-ca21f4173d8cb5ef · revision 1 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-532dcebd6abe5680

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • 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): no

OEIS matches ?

No 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:      1;
  1:      3,       9;
  2:     12,      63,       81;
  3:     45,     351,      891,       729;
  4:    171,    1755,     6642,     10935,      6561;
  5:    648,    8262,    41553,    102789,    124659,      59049;
  6:   2457,   37422,   234738,    774927,   1417176,    1358127,     531441;
  7:   9315,  164997,  1240029,   5119767,  12524949,   18128043,   14348907,    4782969;
  8:  35316,  713205,  6246072,  30956985,  94878621,  183937635,  220016574,  148272039,   43046721;
  9: 133893, 3036285, 30362121, 175605165, 646901478, 1572829164, 2521687545, 2568454353, 1506635235, 387420489;
%F Formula
The row polynomials satisfy P(n,x) = (3+9*x)*P(n-2,x) + (3+9*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 9*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 9*x;
P[n_, x_] := P[n, x] = Expand[(3 + 9*x)*P[n - 2, x] + (3 + 9*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] = 1; v[2] = 3 + 9*x; for(n = 2, N, my(i = n + 1); v[i] = (3 + 9*x)*v[i - 2] + (3 + 9*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, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-45467cdf3192c42e
  • Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
    All coefficients of each polynomial have the same sign (after x -> -x when alternating).
    derived from the cited steps by a stated theorem: e-45467cdf3192c42e
  • Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
    Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
    derived from the cited steps by a stated theorem: e-45467cdf3192c42e

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741