Real-rooted atlas

A159854 → θ+1

nonnegative coefficients only rational roots

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

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
00033
100347
20038516
30031215636
4003163024780
5003205060358176
60032475120105489384
7003281052102451686310832

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-3ba8662663f17943 · revision 1 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-4e3d4a8be3fd75f4

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: 0, 0, 3;
  1: 0, 0, 3,  4;
  2: 0, 0, 3,  8,   5;
  3: 0, 0, 3, 12,  15,   6;
  4: 0, 0, 3, 16,  30,  24,   7;
  5: 0, 0, 3, 20,  50,  60,  35,    8;
  6: 0, 0, 3, 24,  75, 120, 105,   48,   9;
  7: 0, 0, 3, 28, 105, 210, 245,  168,  63,  10;
  8: 0, 0, 3, 32, 140, 336, 490,  448, 252,  80, 11;
  9: 0, 0, 3, 36, 180, 504, 882, 1008, 756, 360, 99, 12;
%F Formula
The row polynomials satisfy (3+(n+2)*x)*P(n,x) = (3+(n+6)*x+(n+3)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 3*x^2.
%t Mathematica
P[0, x_] := 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((3 + (n + 6)*x + (n + 3)*x^2)*P[n - 1, x])/(3 + (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*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((3 + (n + 6)*x + (n + 3)*x^2)*v[i - 1])/(3 + (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-2c18656945c52bb4
  • 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-2c18656945c52bb4
  • 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-2c18656945c52bb4

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 (3)

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

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.