Real-rooted atlas

A063967 → basis A007318 → basis A013610

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2336
31221942
4451179927288
5171585738405811980
6648275446173807153924313608
7245712474260822870117496558972993528
893155499913778118962115462974601196832187642816

All 15 rows

Technical details
Node id
n-9f74ec2fc1b734a2 · revision 0 · JSON
Definition source
derived:basis_map_A013610
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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:      1;
   2:      3,       3;
   3:     12,      21,       9;
   4:     45,     117,      99,      27;
   5:    171,     585,     738,     405,      81;
   6:    648,    2754,    4617,    3807,    1539,     243;
   7:   2457,   12474,   26082,   28701,   17496,    5589,     729;
   8:   9315,   54999,  137781,  189621,  154629,   74601,   19683,    2187;
   9:  35316,  237735,  694008, 1146555, 1171341,  756945,  301806,   67797,   6561;
  10: 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 >= 3, with P(1,x) = 1, P(2,x) = 3 + 3*x.
%t Mathematica
P[1, x_] := 1;
P[2, 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, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + 3*x; for(n = 3, N, my(i = n); 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, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

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
  2. [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