Real-rooted atlas

A113953 → basis A013611

nonnegative coefficients

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

Recurrence
\(P_{n}(x) = \left(2 + 8\,x\right)\,P_{n-2}(x) + \left(1 + 4\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1145
23161635
354411264225
4111204486402561475
52129215043456332810249625
6437044528143362278416384409662875
785162812688513921144321361927782416384410625

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-32899adace0e98a1 · revision 1 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-74aba9cdd78cf583

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 ?

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:   1,    4;
  2:   3,   16,    16;
  3:   5,   44,   112,     64;
  4:  11,  120,   448,    640,     256;
  5:  21,  292,  1504,   3456,    3328,    1024;
  6:  43,  704,  4528,  14336,   22784,   16384,    4096;
  7:  85, 1628, 12688,  51392,  114432,  136192,   77824,   16384;
  8: 171, 3720, 33888, 167040,  480256,  808960,  761856,  360448,   65536;
  9: 341, 8340, 87168, 506880, 1788416, 3917824, 5242880, 4063232, 1638400, 262144;
%F Formula
The row polynomials satisfy P(n,x) = (2+8*x)*P(n-2,x) + (1+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[(2 + 8*x)*P[n - 2, x] + (1 + 4*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] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (2 + 8*x)*v[i - 2] + (1 + 4*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-4cbbf4b06abf62f6
  • 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-4cbbf4b06abf62f6
  • 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-4cbbf4b06abf62f6

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. [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