Real-rooted atlas

A122431 → D

nonnegative coefficients only rational roots

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

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
036312
1169420
202912528
30031215636
400041518744
5000051821852
60000062124960
7000000724271068
80000000827301176
900000000930331284
100000000001033361392
11000000000011363914100
120000000000012394215108
1300000000000013424516116
14000000000000014454817124

Computed in your browser ?

Technical details
Node id
n-59d89a30eaddd429 · revision 1 · JSON
Definition source
derived:derivative
Fingerprint
f6-b6d4a6ea94dc6da0

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: 3, 6, 3;
  1: 1, 6, 9,  4;
  2: 0, 2, 9, 12,  5;
  3: 0, 0, 3, 12, 15,  6;
  4: 0, 0, 0,  4, 15, 18,  7;
  5: 0, 0, 0,  0,  5, 18, 21,  8;
  6: 0, 0, 0,  0,  0,  6, 21, 24,  9;
  7: 0, 0, 0,  0,  0,  0,  7, 24, 27, 10;
  8: 0, 0, 0,  0,  0,  0,  0,  8, 27, 30, 11;
  9: 0, 0, 0,  0,  0,  0,  0,  0,  9, 30, 33, 12;
%F Formula
The row polynomials satisfy (n-1+(n+2)*x)*P(n,x) = (n*x+(n+3)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 3 + 6*x + 3*x^2.
%t Mathematica
P[0, x_] := 3 + 6*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n*x + (n + 3)*x^2)*P[n - 1, x])/(n - 1 + (n + 2)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 3 + 6*x + 3*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((n*x + (n + 3)*x^2)*v[i - 1])/(n - 1 + (n + 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-b524f7ca945d9ab3
  • 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-b524f7ca945d9ab3
  • 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-b524f7ca945d9ab3

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.