Real-rooted atlas

A143685 → basis A007318 → basis A007318

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(18 + 9\,x\right)\,P_{n-2}(x) + \left(3 + x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1314
22715143
3135117271280
48919993513912281
551037209359172951116684
631347527313329196391251631128323
7185895365229282123107649208711917751963760
811219312512863228541510781912795313901527278717319761
9671190316907697177343839996777326130361892165799368199155300564
104033046711276390713386408387700887345159638336115121653910295147791111418835803
11241805655743367861985744323736518177339923223100016613187404032183841152199602112313168458440
121451365371486463376771250294875978691567316688316311006667992531605593809389536530192152717407135123982400441
1387065979033160213906550673430791471901302812822662266311328491601309762524757541500971447103577700129389589371471181477979644
1452244370387204138711243355654666107363985535223242682119307110526048315350842585957788005415119835571512553015587294513897991061115911373436730483
Technical details
Node id
n-70f5dafc292480e1 · revision 0 · JSON
Definition source
derived:basis_map_A007318
Fingerprint
f6-9534464459a0c5cc

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

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:       1;
  1:       3,        1;
  2:      27,       15,        1;
  3:     135,      117,       27,       1;
  4:     891,      999,      351,      39,       1;
  5:    5103,     7209,     3591,     729,      51,      1;
  6:   31347,    52731,    33291,    9639,    1251,     63,     1;
  7:  185895,   365229,   282123,  107649,   20871,   1917,    75,    1;
  8: 1121931,  2512863,  2285415, 1078191,  279531,  39015,  2727,   87,  1;
  9: 6711903, 16907697, 17734383, 9996777, 3261303, 618921, 65799, 3681, 99, 1;
%F Formula
The row polynomials satisfy P(n,x) = (18+9*x)*P(n-2,x) + (3+x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + x;
P[n_, x_] := P[n, x] = Expand[(18 + 9*x)*P[n - 2, x] + (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 + x; for(n = 2, N, my(i = n + 1); v[i] = (18 + 9*x)*v[i - 2] + (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)

polytool c48629374597

Properties

Proof chains ? JSON

Candidate equalities ?

References

  1. [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