Real-rooted atlas

A207606 → 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
\(\left(12 - 66\,x + 12\,x^{2} + 4\,n - 66\,n x + 26\,n x^{2} - 18\,n^{2} x + 18\,n^{2} x^{2} - 2\,n^{3} x + 4\,n^{3} x^{2}\right)\,P_{n}(x) = \left(16 - 64\,x - 68\,x^{2} + 12\,x^{3} + 4\,n - 74\,n x - 32\,n x^{2} + 22\,n x^{3} - 20\,n^{2} x + 7\,n^{2} x^{2} + 12\,n^{2} x^{3} - 2\,n^{3} x + 3\,n^{3} x^{2} + 2\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(8\,x - 86\,x^{2} + 2\,x^{3} + 6\,x^{4} - 28\,n x^{2} + 25\,n x^{3} + 8\,n x^{4} - 4\,n^{2} x^{2} + 7\,n^{2} x^{3} + 2\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 4 + 14\,x + 6\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0414624
153233878
26601086010244
371002732569512743
48154588816500138142218
59224113421601925864189166521
610312201650166050390013722481818942
7114203366105601644514196710520483152054486

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-196be00e036fc9e5 · revision 1 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-bfd7ac197b7c1c0e

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:  4,  14,    6;
  1:  5,  32,   33,     8;
  2:  6,  60,  108,    60,    10;
  3:  7, 100,  273,   256,    95,     12;
  4:  8, 154,  588,   816,   500,    138,     14;
  5:  9, 224, 1134,  2160,  1925,    864,    189,    16;
  6: 10, 312, 2016,  5016,  6050,   3900,   1372,   248,    18;
  7: 11, 420, 3366, 10560, 16445,  14196,   7105,  2048,   315,   20;
  8: 12, 550, 5346, 20592, 40040,  44226,  29400, 11968,  2916,  390,  22;
  9: 13, 704, 8151, 37752, 89375, 122304, 103292, 55488, 18981, 4000, 473, 24;
%F Formula
The row polynomials satisfy (4*n+12-(2*n^3+18*n^2+66*n+66)*x+(4*n^3+18*n^2+26*n+12)*x^2)*P(n,x) = (4*n+16-(2*n^3+20*n^2+74*n+64)*x+(3*n^3+7*n^2-32*n-68)*x^2+(2*n^3+12*n^2+22*n+12)*x^3)*P(n-1,x) + (8*x-(4*n^2+28*n+86)*x^2+(7*n^2+25*n+2)*x^3+(2*n^2+8*n+6)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 4 + 14*x + 6*x^2.
%t Mathematica
P[0, x_] := 4 + 14*x + 6*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((4*n + 16 - (2*n^3 + 20*n^2 + 74*n + 64)*x + (3*n^3 + 7*n^2 - 32*n - 68)*x^2 + (2*n^3 + 12*n^2 + 22*n + 12)*x^3)*P[n - 1, x] + (8*x - (4*n^2 + 28*n + 86)*x^2 + (7*n^2 + 25*n + 2)*x^3 + (2*n^2 + 8*n + 6)*x^4)*D[P[n - 1, x], x])/(4*n + 12 - (2*n^3 + 18*n^2 + 66*n + 66)*x + (4*n^3 + 18*n^2 + 26*n + 12)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 4 + 14*x + 6*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((4*n + 16 - (2*n^3 + 20*n^2 + 74*n + 64)*x + (3*n^3 + 7*n^2 - 32*n - 68)*x^2 + (2*n^3 + 12*n^2 + 22*n + 12)*x^3)*v[i - 1] + (8*x - (4*n^2 + 28*n + 86)*x^2 + (7*n^2 + 25*n + 2)*x^3 + (2*n^2 + 8*n + 6)*x^4)*deriv(v[i - 1], x))/(4*n + 12 - (2*n^3 + 18*n^2 + 66*n + 66)*x + (4*n^3 + 18*n^2 + 26*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 - 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-a214d12805bdd14d
  • 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-a214d12805bdd14d
  • 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-a214d12805bdd14d
  • 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-a214d12805bdd14d

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