Real-rooted atlas

A111125 → (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
\(\left(-12 + 66\,x - 12\,x^{2} + 8\,n - 134\,n x + 44\,n x^{2} + 90\,n^{2} x - 63\,n^{2} x^{2} - 20\,n^{3} x + 44\,n^{3} x^{2} - 15\,n^{4} x^{2} + 2\,n^{5} x^{2}\right)\,P_{n}(x) = \left(-4 + 6\,x + 4\,x^{2} + 8\,n - 58\,n x - 24\,n x^{2} + 70\,n^{2} x + 17\,n^{2} x^{2} - 4\,n^{2} x^{3} - 20\,n^{3} x + 14\,n^{3} x^{2} + 8\,n^{3} x^{3} - 13\,n^{4} x^{2} - 5\,n^{4} x^{3} + 2\,n^{5} x^{2} + n^{5} x^{3}\right)\,P_{n-1}(x) + \left(16\,x - 12\,x^{2} - 4\,x^{3} + 80\,n x^{2} + 20\,n x^{3} - 40\,n^{2} x^{2} + 6\,n^{2} x^{3} + 4\,n^{2} x^{4} - 16\,n^{3} x^{3} - 4\,n^{3} x^{4} + 4\,n^{4} x^{3} + n^{4} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2347
3520934
47566316142
5912024314425541
611220693704275361939
713364163824961625468496653
81556034027200687532407356422091

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-c5edd4429c7197d1 · revision 1 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-d9681d84f8ff1e97

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:  3,    4;
   3:  5,   20,     9;
   4:  7,   56,    63,    16;
   5:  9,  120,   243,   144,    25;
   6: 11,  220,   693,   704,   275,    36;
   7: 13,  364,  1638,  2496,  1625,   468,    49;
   8: 15,  560,  3402,  7200,  6875,  3240,   735,   64;
   9: 17,  816,  6426, 17952, 23375, 15912,  5831, 1088,   81;
  10: 19, 1140, 11286, 40128, 67925, 62244, 32585, 9728, 1539, 100;
%F Formula
The row polynomials satisfy (8*n-12-(20*n^3-90*n^2+134*n-66)*x+(2*n^5-15*n^4+44*n^3-63*n^2+44*n-12)*x^2)*P(n,x) = (8*n-4-(20*n^3-70*n^2+58*n-6)*x+(2*n^5-13*n^4+14*n^3+17*n^2-24*n+4)*x^2+(n^5-5*n^4+8*n^3-4*n^2)*x^3)*P(n-1,x) + (16*x-(40*n^2-80*n+12)*x^2+(4*n^4-16*n^3+6*n^2+20*n-4)*x^3+(n^4-4*n^3+4*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 - 4 - (20*n^3 - 70*n^2 + 58*n - 6)*x + (2*n^5 - 13*n^4 + 14*n^3 + 17*n^2 - 24*n + 4)*x^2 + (n^5 - 5*n^4 + 8*n^3 - 4*n^2)*x^3)*P[n - 1, x] + (16*x - (40*n^2 - 80*n + 12)*x^2 + (4*n^4 - 16*n^3 + 6*n^2 + 20*n - 4)*x^3 + (n^4 - 4*n^3 + 4*n^2)*x^4)*D[P[n - 1, x], x])/(8*n - 12 - (20*n^3 - 90*n^2 + 134*n - 66)*x + (2*n^5 - 15*n^4 + 44*n^3 - 63*n^2 + 44*n - 12)*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 - 4 - (20*n^3 - 70*n^2 + 58*n - 6)*x + (2*n^5 - 13*n^4 + 14*n^3 + 17*n^2 - 24*n + 4)*x^2 + (n^5 - 5*n^4 + 8*n^3 - 4*n^2)*x^3)*v[i - 1] + (16*x - (40*n^2 - 80*n + 12)*x^2 + (4*n^4 - 16*n^3 + 6*n^2 + 20*n - 4)*x^3 + (n^4 - 4*n^3 + 4*n^2)*x^4)*deriv(v[i - 1], x))/(8*n - 12 - (20*n^3 - 90*n^2 + 134*n - 66)*x + (2*n^5 - 15*n^4 + 44*n^3 - 63*n^2 + 44*n - 12)*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-c38cb060e3b23574
  • 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-c38cb060e3b23574
  • 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-c38cb060e3b23574
  • 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-c38cb060e3b23574

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