Real-rooted atlas

A046716 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(2 + 16\,x + 44\,x^{2} + 48\,x^{3} + 18\,x^{4} - n - 8\,n x - 22\,n x^{2} - 24\,n x^{3} - 9\,n x^{4}\right)\,P_{n-2}(x) + \left(-2\,x - 4\,x^{2} - 2\,x^{3} + n + 5\,n x + 7\,n x^{2} + 3\,n x^{3}\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2246
35201944
41698192118424
5655481682220410215520
632635521516431528316461208894304
71957263681458614226726736395560801837272010304
81370022095015105245663438125441561635440211559524338924251255936

All 15 rows

Technical details
Node id
n-dc272a0bfbf3cd6c · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-c788711324b9c3e3

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:      5,       20,        19;
   4:     16,       98,       192,        118;
   5:     65,      548,      1682,       2204,       1021;
   6:    326,     3552,     15164,      31528,      31646,      12088;
   7:   1957,    26368,    145861,     422672,     673639,     556080,      183727;
   8:  13700,   220950,   1510524,    5663438,   12544156,   16354402,    11559524,     3389242;
   9: 109601,  2064280,  16866692,   77987992,  222826414,  401979784,   445815060,   276741704,   73156249;
  10: 986410, 21286428, 202790568, 1118426240, 3930971916, 9118215192, 13932917832, 13490750688, 7485360858, 1804349548;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = -(n-2+(8*n-16)*x+(22*n-44)*x^2+(24*n-48)*x^3+(9*n-18)*x^4)*P(n-2,x) + (n+(5*n-2)*x+(7*n-4)*x^2+(3*n-2)*x^3)*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[(-(n - 2 + (8*n - 16)*x + (22*n - 44)*x^2 + (24*n - 48)*x^3 + (9*n - 18)*x^4)*P[n - 2, x] + (n + (5*n - 2)*x + (7*n - 4)*x^2 + (3*n - 2)*x^3)*P[n - 1, x])/(1 + 2*x + 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] = (-(n - 2 + (8*n - 16)*x + (22*n - 44)*x^2 + (24*n - 48)*x^3 + (9*n - 18)*x^4)*v[i - 2] + (n + (5*n - 2)*x + (7*n - 4)*x^2 + (3*n - 2)*x^3)*v[i - 1])/(1 + 2*x + 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

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