A327997 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

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

\(P_{0}(x) = 1\)

\(P_{1}(x) = 3 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	3	4							7
2	9	25	17						51
3	27	119	169	78					393
4	81	516	1163	1120	393				3273
5	243	2184	7061	10724	7811	2208			30231
6	729	9489	42004	89911	102064	59373	13977		317547
7	2187	44631	262830	742699	1163926	1041025	499437	100026	3856761

All 15 rows

Technical details

Node id: n-6ca0d5556aa7b862 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-0624213b7dc70468

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:     3,       4;
  2:     9,      25,       17;
  3:    27,     119,      169,       78;
  4:    81,     516,     1163,     1120,       393;
  5:   243,    2184,     7061,    10724,      7811,      2208;
  6:   729,    9489,    42004,    89911,    102064,     59373,     13977;
  7:  2187,   44631,   262830,   742699,   1163926,   1041025,    499437,    100026;
  8:  6561,  240480,  1823310,  6466552,  13007259,  15791784,  11515118,   4660200,   806769;
  9: 19683, 1547892, 14439474, 61500032, 150786777, 231118908, 226277794, 138085416, 48009123, 7280604;

%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+2+(2*n+5)*x+(n+3)*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 4*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 4*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 + 2 + (2*n + 5)*x + (n + 3)*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] = 3 + 4*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 + 2 + (2*n + 5)*x + (n + 3)*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

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in [-1, 0]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle read by rows: coefficients of the polynomials given by KummerU(-n, 1 - n - x, 3).?
  - Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

References

[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[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
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023