Real-rooted atlas

A099093 → basis A007318

nonnegative coefficients

Proved for all n: all roots real, in (−∞, −1] · coefficients of one sign · Hadamard kernel what this means

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1336
21221942
3451179927288
4171585738405811980
5648275446173807153924313608
6245712474260822870117496558972993528
793155499913778118962115462974601196832187642816
835316237735694008114655511713417569453018066779765614418064
913389310120953373569650389579864386472548345910511744192296351968330365280
105076274257117159522213515408150424687491618163297121215009381443523676763759049208700064
1119245601773219673785006182951298300207303342136467276194043157842351625460131635657325391071771471434392064
1272965617326450033517962092352781817062121072225790819210139061414460509617194989432523163775916709883259095314419858552768
13276633633006534511499883966454633122693386955781372304008814836535829119444339077157814804315215371899113746521062778327103491159432367757668992
141048797721226633625662694461121914767890495443001878098121577698660272050909852339336380339584533845351526133643239743807774765739765872876877654782969465697330560

Generated b-files: A057089 (row sums) ?

Computed in your browser ?

Other recurrences

X ↦ x+1 substitution · proved equal for all n (proof step)

\(\left(n\right)\,P_{n}(x) = \left(5 + 3\,x - 2\,n\right)\,P_{n-1}(x) + \left(7 + 10\,x + 3\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
Technical details
Node id
n-1f0d56c568ab8221 · revision 2 · JSON
Definition source
derived:basis_map_A007318
derived:shift_x_plus_one (equal by e-e2921515912c97e5)
Fingerprint
f6-c912b16b18733d86

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 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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,       3;
  2:     12,      21,       9;
  3:     45,     117,      99,      27;
  4:    171,     585,     738,     405,      81;
  5:    648,    2754,    4617,    3807,    1539,     243;
  6:   2457,   12474,   26082,   28701,   17496,    5589,     729;
  7:   9315,   54999,  137781,  189621,  154629,   74601,   19683,    2187;
  8:  35316,  237735,  694008, 1146555, 1171341,  756945,  301806,   67797,   6561;
  9: 133893, 1012095, 3373569, 6503895, 7986438, 6472548, 3459105, 1174419, 229635, 19683;
%F Formula
The row polynomials satisfy P(n,x) = (3+3*x)*P(n-2,x) + (3+3*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 3*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 3*x;
P[n_, x_] := P[n, x] = Expand[(3 + 3*x)*P[n - 2, x] + (3 + 3*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 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (3 + 3*x)*v[i - 2] + (3 + 3*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-3252cc2fe5229b3d
  • 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-3252cc2fe5229b3d, e-5bc866dd5a1bb8ee
  • 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-3252cc2fe5229b3d, e-5bc866dd5a1bb8ee
  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-5bc866dd5a1bb8ee

Proof chains

3 incoming proof steps. 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