Real-rooted atlas

A116414 → basis A007318 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, −1/2)coefficients of one signHadamard kernel

Recurrence

\(\left(2\,n + 4\,n x + 2\,n x^{2}\right)\,P_{n}(x) = \left(18 + 78\,x + 126\,x^{2} + 90\,x^{3} + 24\,x^{4} - 8\,n - 46\,n x - 92\,n x^{2} - 78\,n x^{3} - 24\,n x^{4}\right)\,P_{n-1}(x) + \left(13 + 75\,x + 171\,x^{2} + 193\,x^{3} + 108\,x^{4} + 24\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
15611
2225433109
3953544381801067
440920463825316898110429
51760110522767834560215195346101915
675735722817972430027628152914045429133995917
73258528786210873622276820285435021426128917561587609732107

All 15 rows

Technical details
Node id
n-94d1bd0c0d842b87 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-e43ae484f839b1ba

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

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:      1;
  1:      5,       6;
  2:     22,      54,       33;
  3:     95,     354,      438,       180;
  4:    409,    2046,     3825,      3168,       981;
  5:   1760,   11052,    27678,     34560,     21519,      5346;
  6:   7573,   57228,   179724,    300276,    281529,    140454,     29133;
  7:  32585,  287862,  1087362,   2276820,   2854350,   2142612,    891756,    158760;
  8: 140206, 1417698,  6258723,  15757416,  24747255,  24827796,  15539742,   5548176,   865161;
  9: 603275, 6870630, 34712790, 102121200, 192794715, 242238114, 202571496, 108724680, 33987033, 4714686;
%F Formula
The row polynomials satisfy (2*n+4*n*x+2*n*x^2)*P(n,x) = -(8*n-18+(46*n-78)*x+(92*n-126)*x^2+(78*n-90)*x^3+(24*n-24)*x^4)*P(n-1,x) + (13+75*x+171*x^2+193*x^3+108*x^4+24*x^5)*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[(-(8*n - 18 + (46*n - 78)*x + (92*n - 126)*x^2 + (78*n - 90)*x^3 + (24*n - 24)*x^4)*P[n - 1, x] + (13 + 75*x + 171*x^2 + 193*x^3 + 108*x^4 + 24*x^5)*D[P[n - 1, x], x])/(2*n + 4*n*x + 2*n*x^2)]];
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] = (-(8*n - 18 + (46*n - 78)*x + (92*n - 126)*x^2 + (78*n - 90)*x^3 + (24*n - 24)*x^4)*v[i - 1] + (13 + 75*x + 171*x^2 + 193*x^3 + 108*x^4 + 24*x^5)*deriv(v[i - 1], x))/(2*n + 4*n*x + 2*n*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)

polytool c48629374597

Properties

Proof chains ? JSON

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