Real-rooted atlas

A125662 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(5 + 23\,x + 39\,x^{2} + 29\,x^{3} + 8\,x^{4} - n - 9\,n x - 23\,n x^{2} - 23\,n x^{3} - 8\,n x^{4}\right)\,P_{n-1}(x) + \left(3 + 19\,x + 47\,x^{2} + 57\,x^{3} + 34\,x^{4} + 8\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
14610
215463596
356260400204920
420913003018310011898816
57806082188962924422548693084480
629112729010624921991625524915753040391809536
71086411898455689614440402240560208034410703362354167757440

All 15 rows

Technical details
Node id
n-a78dd94efcf1981c · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-b0dd9024715e5b86

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:      4,       6;
  2:     15,      46,       35;
  3:     56,     260,      400,      204;
  4:    209,    1300,     3018,     3100,      1189;
  5:    780,    6082,    18896,    29244,     22548,      6930;
  6:   2911,   27290,   106249,   219916,    255249,    157530,     40391;
  7:  10864,  118984,   556896,  1444040,   2240560,   2080344,   1070336,    235416;
  8:  40545,  508008,  2777748,  8657832,  16825150,  20876792,  16152708,   7125368,  1372105;
  9: 151316, 2134590, 13353312, 48621000, 113562056, 176452564, 182400000, 120961288, 46699460, 7997214;
%F Formula
The row polynomials satisfy (n+2*n*x+n*x^2)*P(n,x) = -(n-5+(9*n-23)*x+(23*n-39)*x^2+(23*n-29)*x^3+(8*n-8)*x^4)*P(n-1,x) + (3+19*x+47*x^2+57*x^3+34*x^4+8*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[(-(n - 5 + (9*n - 23)*x + (23*n - 39)*x^2 + (23*n - 29)*x^3 + (8*n - 8)*x^4)*P[n - 1, x] + (3 + 19*x + 47*x^2 + 57*x^3 + 34*x^4 + 8*x^5)*D[P[n - 1, x], x])/(n + 2*n*x + 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] = (-(n - 5 + (9*n - 23)*x + (23*n - 39)*x^2 + (23*n - 29)*x^3 + (8*n - 8)*x^4)*v[i - 1] + (3 + 19*x + 47*x^2 + 57*x^3 + 34*x^4 + 8*x^5)*deriv(v[i - 1], x))/(n + 2*n*x + 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