Real-rooted atlas

A099093 → basis A013611

nonnegative coefficients

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

Recurrence
\(P_{n}(x) = \left(3 + 12\,x\right)\,P_{n-2}(x) + \left(3 + 12\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 12\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
131215
21284144240
345468158417283825
4171234011808259202073660975
56481101673872243648393984248832972000
6245749896417312183686444789765723136298598415494625
7931521999622044961213574439585024763914248062156835831808246999375
8353169509401110412873379520299863296775111680123619737611107860484299816963937410000
913389340483805397710441624928020445281286627889152141684940801924168089615049359360515978035262766140625
105076271702846825523553622498611841290871987250341699584135050084352245913698304290667626496201231433728619173642241000553259375
11192456070928784118056009611708883072768530695683503477422081131290800128258608907878440990155079684287777472512266244666163274300837068815949791000000
1272965612930580005362873920591057803524367902993922279209798656860729595494423692098945024471530827284486614322433228862041198952448349213934223368916100448256254255163890625
13276633631202613804239981434562909651984642390706067968140523930501126077045075558419569760513228846909455099494482631818425139210392829584998408834369527480324547211228610561069932053790724053074323359375
141048797724906534500106031113776140254514496012683340847872829247649546244041124743168001490702072758272418141935009792088723558304317441401350937536102415970964927938560124112118239723525884626295848960128391846454886464609942308750000

Computed in your browser ?

Technical details
Node id
n-f66027673fc0bfa3 · revision 1 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-6f2ee9b0394095f3

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,      12;
  2:     12,      84,      144;
  3:     45,     468,     1584,      1728;
  4:    171,    2340,    11808,     25920,      20736;
  5:    648,   11016,    73872,    243648,     393984,     248832;
  6:   2457,   49896,   417312,   1836864,    4478976,    5723136,     2985984;
  7:   9315,  219996,  2204496,  12135744,   39585024,   76391424,    80621568,    35831808;
  8:  35316,  950940, 11104128,  73379520,  299863296,  775111680,  1236197376,  1110786048,   429981696;
  9: 133893, 4048380, 53977104, 416249280, 2044528128, 6627889152, 14168494080, 19241680896, 15049359360, 5159780352;
%F Formula
The row polynomials satisfy P(n,x) = (3+12*x)*P(n-2,x) + (3+12*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 12*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 12*x;
P[n_, x_] := P[n, x] = Expand[(3 + 12*x)*P[n - 2, x] + (3 + 12*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 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = (3 + 12*x)*v[i - 2] + (3 + 12*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-8019542a5a1f5666
  • 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-8019542a5a1f5666
  • 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-8019542a5a1f5666

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