Real-rooted atlas

A091042 → D+1

nonnegative coefficients

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

Recurrence
\(\left(-21 + 33\,x - 6\,x^{2} + 35\,n - 34\,n x + 4\,n x^{2} - 20\,n^{2} + 8\,n^{2} x + 4\,n^{3}\right)\,P_{n}(x) = \left(-6 + 21\,x + 75\,x^{2} - 18\,x^{3} + 13\,n + 25\,n x - 86\,n x^{2} + 12\,n x^{3} - 12\,n^{2} - 44\,n^{2} x + 24\,n^{2} x^{2} + 4\,n^{3} + 12\,n^{3} x\right)\,P_{n-1}(x) + \left(2\,x - 30\,x^{2} + 36\,x^{3} - 8\,x^{4} - 16\,n x + 32\,n x^{2} - 16\,n x^{3} + 8\,n^{2} x - 8\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2437
31120536
42291567176
5372883781209832
65671517161122220113840
77915085863686427173641317408
8106283516380307452145057335601577824
9137489639508109616121550565761094881617344064
10172792385272329460537472394706131784193801140191507328
1121112180168777868224196707021162961107890279072323191540216553600
1225417963311696206220362106449256534707240827784465479985135920242328311552
13301256005439504503400174253753447056036402100208012006369275101200078430260025121634816
143523545190558091763104440150011273944515779196012436146055219320135671251776060115830327627520093696
15407479081448811176436001044421953314056205948072556204700803792395251347473402718358529858401662574060292214592512

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Technical details
Node id
n-4e46f9592d8d818b · revision 1 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-5e0ca6f4c7830ccb

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:   4,    3;
   3:  11,   20,     5;
   4:  22,   91,    56,      7;
   5:  37,  288,   378,    120,      9;
   6:  56,  715,  1716,   1122,    220,     11;
   7:  79, 1508,  5863,   6864,   2717,    364,     13;
   8: 106, 2835, 16380,  30745,  21450,   5733,    560,    15;
   9: 137, 4896, 39508, 109616, 121550,  56576,  10948,   816,   17;
  10: 172, 7923, 85272, 329460, 537472, 394706, 131784, 19380, 1140, 19;
%F Formula
The row polynomials satisfy (4*n^3-20*n^2+35*n-21+(8*n^2-34*n+33)*x+(4*n-6)*x^2)*P(n,x) = (4*n^3-12*n^2+13*n-6+(12*n^3-44*n^2+25*n+21)*x+(24*n^2-86*n+75)*x^2+(12*n-18)*x^3)*P(n-1,x) + ((8*n^2-16*n+2)*x-(8*n^2-32*n+30)*x^2-(16*n-36)*x^3-8*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[((4*n^3 - 12*n^2 + 13*n - 6 + (12*n^3 - 44*n^2 + 25*n + 21)*x + (24*n^2 - 86*n + 75)*x^2 + (12*n - 18)*x^3)*P[n - 1, x] + ((8*n^2 - 16*n + 2)*x - (8*n^2 - 32*n + 30)*x^2 - (16*n - 36)*x^3 - 8*x^4)*D[P[n - 1, x], x])/(4*n^3 - 20*n^2 + 35*n - 21 + (8*n^2 - 34*n + 33)*x + (4*n - 6)*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] = ((4*n^3 - 12*n^2 + 13*n - 6 + (12*n^3 - 44*n^2 + 25*n + 21)*x + (24*n^2 - 86*n + 75)*x^2 + (12*n - 18)*x^3)*v[i - 1] + ((8*n^2 - 16*n + 2)*x - (8*n^2 - 32*n + 30)*x^2 - (16*n - 36)*x^3 - 8*x^4)*deriv(v[i - 1], x))/(4*n^3 - 20*n^2 + 35*n - 21 + (8*n^2 - 34*n + 33)*x + (4*n - 6)*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-c924eee3366ab2fb
  • 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-c924eee3366ab2fb
  • 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-c924eee3366ab2fb
  • 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-c924eee3366ab2fb

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