Real-rooted atlas

A099093 → basis A013609

nonnegative coefficients

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

Recurrence
\(P_{n}(x) = \left(3 + 6\,x\right)\,P_{n-2}(x) + \left(3 + 6\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 6\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1369
212423690
345234396216891
417111702952324012968829
56485508184683045624624777687480
624572494810432822960827993617884846656866781
7931510999855112415169682474064238723212597122799368588349
835316475470277603291724401874145624222240193155848678016167961685096170
9133893202419013494276520311601277830082071215362213827201503256325878656010077696843160671
105076278514234638088842812326488067949921573178112211015756819212007681135420416393030144604661768354311569
1119245603546439229514002414636103844803316848109483669441767641875220203820928160117793288374565376260004556836279705682777250160
12729656114652900013407184807388222544272993937127122530620813448899929618509452300818419172940812918598502460587108352170514616322176782336820184055561
13276633636013069025999535864363706498081494191292484391372828169495382930561528887540096183240058982416139027036161014924764160431365699584111015899136130606940168126651751489
141048797722453267250265077784441753181431207927088029922591398904832631425741120011646109943424163336693363201732881998131213685067749376779832271872030300810117127183381708807836416409680521522263450

Generated b-files: A057092 (row sums) ?

Computed in your browser ?

Technical details
Node id
n-f8c44f01f6cd4a34 · revision 1 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-a264a61b871c93de

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,       6;
  2:     12,      42,       36;
  3:     45,     234,      396,      216;
  4:    171,    1170,     2952,     3240,      1296;
  5:    648,    5508,    18468,    30456,     24624,      7776;
  6:   2457,   24948,   104328,   229608,    279936,    178848,     46656;
  7:   9315,  109998,   551124,  1516968,   2474064,   2387232,   1259712,    279936;
  8:  35316,  475470,  2776032,  9172440,  18741456,  24222240,  19315584,   8678016,  1679616;
  9: 133893, 2024190, 13494276, 52031160, 127783008, 207121536, 221382720, 150325632, 58786560, 10077696;
%F Formula
The row polynomials satisfy P(n,x) = (3+6*x)*P(n-2,x) + (3+6*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 6*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 6*x;
P[n_, x_] := P[n, x] = Expand[(3 + 6*x)*P[n - 2, x] + (3 + 6*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 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = (3 + 6*x)*v[i - 2] + (3 + 6*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-8e828a8330d7037f
  • 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-8e828a8330d7037f
  • 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-8e828a8330d7037f

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