Real-rooted atlas

A327997 → x ↦ x+1 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(3 + 9\,x + 9\,x^{2} + 3\,x^{3} - 3\,n - 9\,n x - 9\,n x^{2} - 3\,n x^{3}\right)\,P_{n-2}(x) + \left(3 + 7\,x + 4\,x^{2} + n + 2\,n x + n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 5\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1459
217432787
378299379159915
4393202438753272102910593
52208142693655646459293167353136161
613977108351346954588001556714279387580951951479
71000269009273449965728738491769316893732286181150669131177467

All 15 rows

Technical details
Node id
n-e644e56dcd6039c0 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-3da5096eb0595747

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,        5;
  2:      17,       43,        27;
  3:      78,      299,       379,        159;
  4:     393,     2024,      3875,       3272,       1029;
  5:    2208,    14269,     36556,      46459,      29316,       7353;
  6:   13977,   108351,    346954,     588001,     556714,     279387,      58095;
  7:  100026,   900927,   3449965,    7287384,    9176931,    6893732,    2861811,     506691;
  8:  806769,  8248104,  36630446,   92365344,  144721779,  144357088,   89559462,   31608216,   4860297;
  9: 7280604, 83041749, 418346424, 1222462142, 2284570884, 2832828711, 2331526744, 1228594518, 376225740, 51023277;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = -(3*n-3+(9*n-9)*x+(9*n-9)*x^2+(3*n-3)*x^3)*P(n-2,x) + (n+3+(2*n+7)*x+(n+4)*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 5*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 5*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(3*n - 3 + (9*n - 9)*x + (9*n - 9)*x^2 + (3*n - 3)*x^3)*P[n - 2, x] + (n + 3 + (2*n + 7)*x + (n + 4)*x^2)*P[n - 1, x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 5*x; for(n = 2, N, my(i = n + 1); v[i] = (-(3*n - 3 + (9*n - 9)*x + (9*n - 9)*x^2 + (3*n - 3)*x^3)*v[i - 2] + (n + 3 + (2*n + 7)*x + (n + 4)*x^2)*v[i - 1])/(1 + x)); 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. [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