Real-rooted atlas

A081579 → x ↦ x+1 → basis A013610

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(8 + 12\,x\right)\,P_{n-2}(x) + \left(3 + 3\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 3\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1336
21730956
37520115327456
436112721494648813856
516837407119348478251124332256
67937417788523984348413919234729270656
7372752285975654077197934990411839513280521872269056

All 15 rows

Technical details
Node id
n-2bfc2c973ebf670c · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-cd842626bf69cd22

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,       3;
  2:     17,      30,        9;
  3:     75,     201,      153,       27;
  4:    361,    1272,     1494,      648,       81;
  5:   1683,    7407,    11934,     8478,     2511,      243;
  6:   7937,   41778,    85239,    84348,    41391,     9234,      729;
  7:  37275,  228597,   565407,   719793,   499041,   183951,    32805,    2187;
  8: 175321, 1227084,  3565260,  5553252,  4999806,  2619540,   766908,  113724,   6561;
  9: 824163, 6483291, 21643452, 39898764, 44289018, 30318138, 12629196, 3053052, 387099, 19683;
%F Formula
The row polynomials satisfy P(n,x) = (8+12*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[(8 + 12*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] = (8 + 12*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)

polytool c48629374597

Properties

Proof chains ? JSON

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