Real-rooted atlas

A085478 → x ↦ x+1

nonnegative coefficients

Proved for all n: all roots real, in (−5, −1) · coefficients of one sign · interlacing (weak) · Hadamard kernel what this means

Recurrence
\(\left(-3 + 2\,n\right)\,P_{n}(x) = \left(-4 - x + 3\,n + n x\right)\,P_{n-1}(x) + \left(5 + 6\,x + x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
011
111
2213
355111
413198141
5346542111153
68921018374141571
72336547173941151712131

Show all 15 stored rows

Generated b-files: A001835 (row sums), A079935 (row sums) ?

Computed in your browser ?

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(P_{n}(x) = \left(-1\right)\,P_{n-2}(x) + \left(3 + x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)
Technical details
Node id
n-d0986a7acf60271a · revision 1 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-a08c2f082979b173)
Fingerprint
f6-525575d96ecebcc3

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
  • 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;
  2:    2,    1;
  3:    5,    5,    1;
  4:   13,   19,    8,    1;
  5:   34,   65,   42,   11,    1;
  6:   89,  210,  183,   74,   14,    1;
  7:  233,  654,  717,  394,  115,   17,   1;
  8:  610, 1985, 2622, 1825,  725,  165,  20,  1;
  9: 1597, 5911, 9134, 7703, 3885, 1203, 224, 23, 1;
%F Formula
The row polynomials satisfy (2*n-3)*P(n,x) = (3*n-4+(n-1)*x)*P(n-1,x) + (5+6*x+x^2)*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[((3*n - 4 + (n - 1)*x)*P[n - 1, x] + (5 + 6*x + x^2)*D[P[n - 1, x], x])/(2*n - 3)]];
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] = ((3*n - 4 + (n - 1)*x)*v[i - 1] + (5 + 6*x + x^2)*deriv(v[i - 1], x))/(2*n - 3)); 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 (-5, -1)
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-b3424874909a70f6
  • 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-b3424874909a70f6, e-c1b3c12b9e216f54
  • 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-b3424874909a70f6, e-c1b3c12b9e216f54
  • 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-6f13642fbb28ea1d, e-9f91faf8953a1d7a, e-b3424874909a70f6, e-c1b3c12b9e216f54
  • Real-rooted. Every row P_n, n >= 1, is zero or real-rooted with all roots in (-5, -1)
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-6f13642fbb28ea1d
  • 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-6f13642fbb28ea1d, e-9f91faf8953a1d7a
  • 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-6f13642fbb28ea1d, e-9f91faf8953a1d7a
  • 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-c1b3c12b9e216f54
  • 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-9f91faf8953a1d7a

Proof chains

5 incoming proof steps. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

References

  1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023