Real-rooted atlas

A249250 → θ+1

nonnegative coefficients

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

Recurrence
\(\left(-2 - 8\,x - 14\,x^{2} - 3\,n - 10\,n x + 2\,n x^{2} - n x^{3} - n^{2} - 9\,n^{2} x + 9\,n^{2} x^{2} + n^{2} x^{3} - 3\,n^{3} x + 3\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-2 - 8\,x - 14\,x^{2} - 3\,n - 34\,n x - 85\,n x^{2} + 2\,n x^{3} - n^{2} - 21\,n^{2} x - 66\,n^{2} x^{2} - 2\,n^{2} x^{3} - 3\,n^{3} x - 15\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(2\,x + 35\,x^{2} + 153\,x^{3} + 25\,x^{4} + x^{5} + n x + 23\,n x^{2} + 132\,n x^{3} + 23\,n x^{4} + n x^{5} + 3\,n^{2} x^{2} + 30\,n^{2} x^{3} + 3\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1167
2112922
3118451276
41241089615244
513019836016518772
6136315912855252212392
7142459186028951674357247312
814863033127500720028984802722096
9154828537616290231481520446086213066160
101601053816031290606965901028608688578033196576
111661305117725493013734018383413080049437981095736580288
121721584163209004527835248568847577626095580040134641152391703488
13178189021912139875518238113089214510401090503479850123090179281365424976704
1418422232865620806590219623849913853632379341922714208270131815842328315964514478208

Computed in your browser ?

Technical details
Node id
n-f6acc7b818ac622f · revision 1 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-c3e1b34dd6afb3e0

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 ?

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,  6;
  2: 1, 12,   9;
  3: 1, 18,  45,   12;
  4: 1, 24, 108,   96,    15;
  5: 1, 30, 198,  360,   165,    18;
  6: 1, 36, 315,  912,   855,   252,    21;
  7: 1, 42, 459, 1860,  2895,  1674,   357,   24;
  8: 1, 48, 630, 3312,  7500,  7200,  2898,  480,  27;
  9: 1, 54, 828, 5376, 16290, 23148, 15204, 4608, 621, 30;
%F Formula
The row polynomials satisfy -(n^2+3*n+2+(3*n^3+9*n^2+10*n+8)*x-(3*n^3+9*n^2+2*n-14)*x^2-(n^2-n)*x^3)*P(n,x) = -(n^2+3*n+2+(3*n^3+21*n^2+34*n+8)*x+(15*n^3+66*n^2+85*n+14)*x^2+(2*n^2-2*n)*x^3)*P(n-1,x) + ((n+2)*x+(3*n^2+23*n+35)*x^2+(30*n^2+132*n+153)*x^3+(3*n^2+23*n+25)*x^4+(n+1)*x^5)*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[(-(n^2 + 3*n + 2 + (3*n^3 + 21*n^2 + 34*n + 8)*x + (15*n^3 + 66*n^2 + 85*n + 14)*x^2 + (2*n^2 - 2*n)*x^3)*P[n - 1, x] + ((n + 2)*x + (3*n^2 + 23*n + 35)*x^2 + (30*n^2 + 132*n + 153)*x^3 + (3*n^2 + 23*n + 25)*x^4 + (n + 1)*x^5)*D[P[n - 1, x], x])/(-n^2 - 3*n - 2 - (3*n^3 + 9*n^2 + 10*n + 8)*x + (3*n^3 + 9*n^2 + 2*n - 14)*x^2 + (n^2 - n)*x^3)]];
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] = (-(n^2 + 3*n + 2 + (3*n^3 + 21*n^2 + 34*n + 8)*x + (15*n^3 + 66*n^2 + 85*n + 14)*x^2 + (2*n^2 - 2*n)*x^3)*v[i - 1] + ((n + 2)*x + (3*n^2 + 23*n + 35)*x^2 + (30*n^2 + 132*n + 153)*x^3 + (3*n^2 + 23*n + 25)*x^4 + (n + 1)*x^5)*deriv(v[i - 1], x))/(-n^2 - 3*n - 2 - (3*n^3 + 9*n^2 + 10*n + 8)*x + (3*n^3 + 9*n^2 + 2*n - 14)*x^2 + (n^2 - n)*x^3)); 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-195b29f840e130c6
  • 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-195b29f840e130c6
  • 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-195b29f840e130c6
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-195b29f840e130c6

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