Real-rooted atlas

A033282 → (θ+1)²

nonnegative coefficients

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

Recurrence
\(\left(-2\,n - n^{2} + 2\,n^{2} x + 3\,n^{3} x + n^{4} x\right)\,P_{n}(x) = \left(-8\,x + 4\,x^{2} - 2\,n - 12\,n x + 14\,n x^{2} - n^{2} - 2\,n^{2} x + 18\,n^{2} x^{2} + 3\,n^{3} x + 10\,n^{3} x^{2} + n^{4} x + 2\,n^{4} x^{2}\right)\,P_{n-1}(x) + \left(-2\,x + 2\,x^{3} - 2\,n x + 4\,n x^{2} + 6\,n x^{3} + 6\,n^{2} x^{2} + 6\,n^{2} x^{3} + 2\,n^{3} x^{2} + 2\,n^{3} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1189
21204566
3136189224450
4156504134410502955
5180108048008250475218963
61108202513200371254633221021119812
7114034653080012512525225224524591520748548

Show all 15 stored rows

Generated b-files: A002695 (row sums) ?

Computed in your browser ?

Technical details
Node id
n-968b28a80a14d56e · revision 1 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-3284531df95da94f

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 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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,   8;
  2: 1,  20,   45;
  3: 1,  36,  189,    224;
  4: 1,  56,  504,   1344,   1050;
  5: 1,  80, 1080,   4800,   8250,    4752;
  6: 1, 108, 2025,  13200,  37125,   46332,   21021;
  7: 1, 140, 3465,  30800, 125125,  252252,  245245,   91520;
  8: 1, 176, 5544,  64064, 350350, 1009008, 1569568, 1244672,  393822;
  9: 1, 216, 8424, 122304, 859950, 3302208, 7277088, 9165312, 6122142, 1679600;
%F Formula
The row polynomials satisfy -(n^2+2*n-(n^4+3*n^3+2*n^2)*x)*P(n,x) = -(n^2+2*n-(n^4+3*n^3-2*n^2-12*n-8)*x-(2*n^4+10*n^3+18*n^2+14*n+4)*x^2)*P(n-1,x) - ((2*n+2)*x-(2*n^3+6*n^2+4*n)*x^2-(2*n^3+6*n^2+6*n+2)*x^3)*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 + 2*n - (n^4 + 3*n^3 - 2*n^2 - 12*n - 8)*x - (2*n^4 + 10*n^3 + 18*n^2 + 14*n + 4)*x^2)*P[n - 1, x] - ((2*n + 2)*x - (2*n^3 + 6*n^2 + 4*n)*x^2 - (2*n^3 + 6*n^2 + 6*n + 2)*x^3)*D[P[n - 1, x], x])/(-n^2 - 2*n + (n^4 + 3*n^3 + 2*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(n^2 + 2*n - (n^4 + 3*n^3 - 2*n^2 - 12*n - 8)*x - (2*n^4 + 10*n^3 + 18*n^2 + 14*n + 4)*x^2)*v[i - 1] - ((2*n + 2)*x - (2*n^3 + 6*n^2 + 4*n)*x^2 - (2*n^3 + 6*n^2 + 6*n + 2)*x^3)*deriv(v[i - 1], x))/(-n^2 - 2*n + (n^4 + 3*n^3 + 2*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-5812fcfa63c69609
  • 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-5812fcfa63c69609
  • 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-5812fcfa63c69609
  • 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-5812fcfa63c69609

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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023