Real-rooted atlas

A274498 → basis A008277

nonnegative coefficients

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

Recurrence
\(P_{n}(x) = \left(9 + 6\,x\right)\,P_{n-2}(x) + \left(6\,x\right)\,P_{n-2}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3\)

Rows ?

nx0x1x2x3Pn(1)
011
133
29615
3271845
48114436261
5243432108783
6729264616202165211
721877938486064815633

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-944028b132f80faf · revision 1 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-6f5d4412c18f6044

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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 ?

No OEIS matches.

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;
  2:     9,      6;
  3:    27,     18;
  4:    81,    144,     36;
  5:   243,    432,    108;
  6:   729,   2646,   1620,   216;
  7:  2187,   7938,   4860,   648;
  8:  6561,  44064,  49896, 15552, 1296;
  9: 19683, 132192, 149688, 46656, 3888;
%F Formula
The row polynomials satisfy P(n,x) = (9+6*x)*P(n-2,x) + 6*x*d/dx P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[(9 + 6*x)*P[n - 2, x] + 6*x*D[P[n - 2, x], 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; for(n = 2, N, my(i = n + 1); v[i] = (9 + 6*x)*v[i - 2] + 6*x*deriv(v[i - 2], x)); 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-81aaf3fee759d079
  • 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-81aaf3fee759d079
  • 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-81aaf3fee759d079

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

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

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