Real-rooted atlas

A159854 → basis A013610

nonnegative coefficients only rational roots

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

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
016916
1215362780
243611716281400
38843426756482432000
41619293623763321243072910000
5324322448756013770148238748218750000
664960619222464502207095661965306186561250000
712821121526463504167832292572336798247131104976196831250000

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-ca8f77e380f6014f · revision 1 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-60d64fb2105f3f8c

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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): yes

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,    6,     9;
  1:   2,   15,    36,     27;
  2:   4,   36,   117,    162,      81;
  3:   8,   84,   342,    675,     648,     243;
  4:  16,  192,   936,   2376,    3321,    2430,     729;
  5:  32,  432,  2448,   7560,   13770,   14823,    8748,    2187;
  6:  64,  960,  6192,  22464,   50220,   70956,   61965,   30618,    6561;
  7: 128, 2112, 15264,  63504,  167832,  292572,  336798,  247131,  104976,   19683;
  8: 256, 4608, 36864, 172800,  526176, 1088640, 1551312, 1504656,  951345,  354294,   59049;
  9: 512, 9984, 87552, 456192, 1570752, 3755808, 6368544, 7663248, 6416658, 3562623, 1180980, 177147;
%F Formula
The row polynomials satisfy P(n,x) = (2+3*x)*P(n-1,x) for n >= 1, with P(0,x) = 1 + 6*x + 9*x^2.
%t Mathematica
P[0, x_] := 1 + 6*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[(2 + 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 + 6*x + 9*x^2; for(n = 1, N, my(i = n + 1); 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)

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-02e979b2d9fb24b7
  • 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-02e979b2d9fb24b7
  • 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-02e979b2d9fb24b7

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.