Real-rooted atlas

A060920 → reverse

nonnegative coefficients

Proved for all n: all roots real what this means

Recurrence
\(\left(-1 + 2\,n\right)\,P_{n}(x) = \left(-1 + 3\,x + 5\,x^{2} + 2\,n - n x - 5\,n x^{2}\right)\,P_{n-1}(x) + \left(4\,x + 9\,x^{2} + 5\,x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1123
215511
319201343
4114517134171
512010523323589683
61271905949427442332731
7135315129528603522228561010923

Show all 15 stored rows

Generated b-files: A007583 (row sums), A060920 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-80f3c43c239c22ff · revision 1 · JSON
Definition source
derived:reversal
Fingerprint
f6-5870c1c52d074a69

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 0)
  • 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,  5,   5;
  3: 1,  9,  20,   13;
  4: 1, 14,  51,   71,    34;
  5: 1, 20, 105,  233,   235,    89;
  6: 1, 27, 190,  594,   942,   744,   233;
  7: 1, 35, 315, 1295,  2860,  3522,  2285,   610;
  8: 1, 44, 490, 2534,  7285, 12402, 12473,  6865,  1597;
  9: 1, 54, 726, 4578, 16407, 36122, 49963, 42447, 20284, 4181;
%F Formula
The row polynomials satisfy (2*n-1)*P(n,x) = (2*n-1-(n-3)*x-(5*n-5)*x^2)*P(n-1,x) + (4*x+9*x^2+5*x^3)*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[((2*n - 1 - (n - 3)*x - (5*n - 5)*x^2)*P[n - 1, x] + (4*x + 9*x^2 + 5*x^3)*D[P[n - 1, x], x])/(2*n - 1)]];
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] = ((2*n - 1 - (n - 3)*x - (5*n - 5)*x^2)*v[i - 1] + (4*x + 9*x^2 + 5*x^3)*deriv(v[i - 1], x))/(2*n - 1)); 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.

A060920: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:    1;
  1:    2,     1;
  2:    5,     5,     1;
  3:   13,    20,     9,     1;
  4:   34,    71,    51,    14,     1;
  5:   89,   235,   233,   105,    20,     1;
  6:  233,   744,   942,   594,   190,    27,    1;
  7:  610,  2285,  3522,  2860,  1295,   315,   35,   1;
  8: 1597,  6865, 12473, 12402,  7285,  2534,  490,  44,  1;
  9: 4181, 20284, 42447, 49963, 36122, 16407, 4578, 726, 54, 1;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where (2*n-1)*P(n,x) = (2*n-1-(n-3)*x-(5*n-5)*x^2)*P(n-1,x) + (4*x+9*x^2+5*x^3)*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[((2*n - 1 - (n - 3)*x - (5*n - 5)*x^2)*P[n - 1, x] + (4*x + 9*x^2 + 5*x^3)*D[P[n - 1, x], x])/(2*n - 1)]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {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] = ((2*n - 1 - (n - 3)*x - (5*n - 5)*x^2)*v[i - 1] + (4*x + 9*x^2 + 5*x^3)*deriv(v[i - 1], x))/(2*n - 1)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); 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, +inf)
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-2315075e9c2c499e

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

Same fingerprint (3)

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