Real-rooted atlas

A086645 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(-2 - 4\,x - 2\,x^{2} + n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(-8 - 40\,x - 68\,x^{2} - 48\,x^{3} - 12\,x^{4} + 4\,n + 20\,n x + 34\,n x^{2} + 24\,n x^{3} + 6\,n x^{4}\right)\,P_{n-1}(x) + \left(-2\,x - 12\,x^{2} - 24\,x^{3} - 20\,x^{4} - 6\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2246
38322868
432192360208792
512810242944358415529232
6512512019840371203344011584107616
7204824576119808303104418944299520864641254464
88192114688673792215040040212484401152260825664537614623104

All 15 rows

Technical details
Node id
n-3aa1e02b394f91f2 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-3661b733ab1f9507

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

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:      1;
   2:      2,       4;
   3:      8,      32,       28;
   4:     32,     192,      360,      208;
   5:    128,    1024,     2944,     3584,      1552;
   6:    512,    5120,    19840,    37120,     33440,     11584;
   7:   2048,   24576,   119808,   303104,    418944,    299520,     86464;
   8:   8192,  114688,   673792,  2150400,   4021248,   4401152,   2608256,    645376;
   9:  32768,  524288,  3604480, 13893632,  32808960,  48562176,  43966464,  22249472,   4817152;
  10: 131072, 2359296, 18579456, 83951616, 239689728, 448118784, 548278272, 423124992, 186831360, 35955712;
%F Formula
The row polynomials satisfy (n-2+(2*n-4)*x+(n-2)*x^2)*P(n,x) = (4*n-8+(20*n-40)*x+(34*n-68)*x^2+(24*n-48)*x^3+(6*n-12)*x^4)*P(n-1,x) - (2*x+12*x^2+24*x^3+20*x^4+6*x^5)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 4*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n - 8 + (20*n - 40)*x + (34*n - 68)*x^2 + (24*n - 48)*x^3 + (6*n - 12)*x^4)*P[n - 1, x] - (2*x + 12*x^2 + 24*x^3 + 20*x^4 + 6*x^5)*D[P[n - 1, x], x])/(n - 2 + (2*n - 4)*x + (n - 2)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 4*x; for(n = 3, N, my(i = n); v[i] = ((4*n - 8 + (20*n - 40)*x + (34*n - 68)*x^2 + (24*n - 48)*x^3 + (6*n - 12)*x^4)*v[i - 1] - (2*x + 12*x^2 + 24*x^3 + 20*x^4 + 6*x^5)*deriv(v[i - 1], x))/(n - 2 + (2*n - 4)*x + (n - 2)*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)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

References

  1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  3. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023