Real-rooted atlas

A152198 → 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
\(\left(8 + 6\,x + 4\,n x + n x^{2} + n^{2} x^{2}\right)\,P_{n}(x) = \left(8 - 10\,x - 12\,x^{2} - 2\,x^{3} + 20\,n x + 5\,n x^{2} - n x^{3} + 9\,n^{2} x^{2} + 2\,n^{2} x^{3} + n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-16\,x - 32\,x^{2} - 20\,x^{3} - 4\,x^{4} - 12\,n x^{2} - 18\,n x^{3} - 6\,n x^{4} - 2\,n^{2} x^{3} - 2\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
111
2123
3123
41438
51438
6169420
7169420
8181816548
9181816548
101103040256112
111103040256112
12112458075367256
13112458075367256
1411463140175126498576

Computed in your browser ?

Technical details
Node id
n-b6f36b24716e535e · revision 1 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-b1032d52426bb000

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): 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;
  1: 1;
  2: 1, 2;
  3: 1, 2;
  4: 1, 4,  3;
  5: 1, 4,  3;
  6: 1, 6,  9,  4;
  7: 1, 6,  9,  4;
  8: 1, 8, 18, 16, 5;
  9: 1, 8, 18, 16, 5;
%F Formula
The row polynomials satisfy (8+(4*n+6)*x+(n^2+n)*x^2)*P(n,x) = (8+(20*n-10)*x+(9*n^2+5*n-12)*x^2+(n^3+2*n^2-n-2)*x^3)*P(n-1,x) - (16*x+(12*n+32)*x^2+(2*n^2+18*n+20)*x^3+(2*n^2+6*n+4)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((8 + (20*n - 10)*x + (9*n^2 + 5*n - 12)*x^2 + (n^3 + 2*n^2 - n - 2)*x^3)*P[n - 1, x] - (16*x + (12*n + 32)*x^2 + (2*n^2 + 18*n + 20)*x^3 + (2*n^2 + 6*n + 4)*x^4)*D[P[n - 1, x], x])/(8 + (4*n + 6)*x + (n^2 + n)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((8 + (20*n - 10)*x + (9*n^2 + 5*n - 12)*x^2 + (n^3 + 2*n^2 - n - 2)*x^3)*v[i - 1] - (16*x + (12*n + 32)*x^2 + (2*n^2 + 18*n + 20)*x^3 + (2*n^2 + 6*n + 4)*x^4)*deriv(v[i - 1], x))/(8 + (4*n + 6)*x + (n^2 + n)*x^2)); 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-62712304e6278432
  • 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-62712304e6278432
  • 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-62712304e6278432

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.