Real-rooted atlas

A078812 → (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
\(\left(-8 + 20\,x - 2\,x^{2} + 8\,n - 60\,n x + 10\,n x^{2} + 60\,n^{2} x - 20\,n^{2} x^{2} - 20\,n^{3} x + 20\,n^{3} x^{2} - 10\,n^{4} x^{2} + 2\,n^{5} x^{2}\right)\,P_{n}(x) = \left(8\,n - 4\,n x - 2\,n x^{2} + 40\,n^{2} x + 10\,n^{2} x^{2} - 20\,n^{3} x - 6\,n^{3} x^{2} + n^{3} x^{3} - 8\,n^{4} x^{2} - 2\,n^{4} x^{3} + 2\,n^{5} x^{2} + n^{5} x^{3}\right)\,P_{n-1}(x) + \left(16\,x - 4\,x^{2} - 2\,x^{3} + 40\,n x^{2} + 10\,n x^{3} - 40\,n^{2} x^{2} - 6\,n^{2} x^{3} + n^{2} x^{4} - 8\,n^{3} x^{3} - 2\,n^{3} x^{4} + 4\,n^{4} x^{3} + n^{4} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2246
3316928
44405416114
558018912825427
66140504576250361512
77224113419201375432495141
8833622685280550028086866416950

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-ffd64b15cb7f95af · revision 1 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-b5575767b5ed4a03

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:  1;
   2:  2,   4;
   3:  3,  16,    9;
   4:  4,  40,   54,    16;
   5:  5,  80,  189,   128,    25;
   6:  6, 140,  504,   576,   250,    36;
   7:  7, 224, 1134,  1920,  1375,   432,    49;
   8:  8, 336, 2268,  5280,  5500,  2808,   686,   64;
   9:  9, 480, 4158, 12672, 17875, 13104,  5145, 1024,   81;
  10: 10, 660, 7128, 27456, 50050, 49140, 27440, 8704, 1458, 100;
%F Formula
The row polynomials satisfy (8*n-8-(20*n^3-60*n^2+60*n-20)*x+(2*n^5-10*n^4+20*n^3-20*n^2+10*n-2)*x^2)*P(n,x) = (8*n-(20*n^3-40*n^2+4*n)*x+(2*n^5-8*n^4-6*n^3+10*n^2-2*n)*x^2+(n^5-2*n^4+n^3)*x^3)*P(n-1,x) + (16*x-(40*n^2-40*n+4)*x^2+(4*n^4-8*n^3-6*n^2+10*n-2)*x^3+(n^4-2*n^3+n^2)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((8*n - (20*n^3 - 40*n^2 + 4*n)*x + (2*n^5 - 8*n^4 - 6*n^3 + 10*n^2 - 2*n)*x^2 + (n^5 - 2*n^4 + n^3)*x^3)*P[n - 1, x] + (16*x - (40*n^2 - 40*n + 4)*x^2 + (4*n^4 - 8*n^3 - 6*n^2 + 10*n - 2)*x^3 + (n^4 - 2*n^3 + n^2)*x^4)*D[P[n - 1, x], x])/(8*n - 8 - (20*n^3 - 60*n^2 + 60*n - 20)*x + (2*n^5 - 10*n^4 + 20*n^3 - 20*n^2 + 10*n - 2)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((8*n - (20*n^3 - 40*n^2 + 4*n)*x + (2*n^5 - 8*n^4 - 6*n^3 + 10*n^2 - 2*n)*x^2 + (n^5 - 2*n^4 + n^3)*x^3)*v[i - 1] + (16*x - (40*n^2 - 40*n + 4)*x^2 + (4*n^4 - 8*n^3 - 6*n^2 + 10*n - 2)*x^3 + (n^4 - 2*n^3 + n^2)*x^4)*deriv(v[i - 1], x))/(8*n - 8 - (20*n^3 - 60*n^2 + 60*n - 20)*x + (2*n^5 - 10*n^4 + 20*n^3 - 20*n^2 + 10*n - 2)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 1, 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-67a2e8580e66d57b
  • Coefficients of one sign. Every nonzero row, n >= 1: 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-67a2e8580e66d57b
  • Hadamard kernel (Malo). Rows P_n, n >= 1, 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-67a2e8580e66d57b
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-67a2e8580e66d57b

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