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	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	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
8	6561	240480	1823310	6466552	13007259	15791784	11515118	4660200	806769							54318033
9	19683	1547892	14439474	61500032	150786777	231118908	226277794	138085416	48009123	7280604						879065703
10	59049	11983329	131003847	646164942	1859322569	3430287939	4205018817	3424616374	1786878135	542094921	72865089					16110295011
11	177147	108992223	1344168387	7476391944	24570713135	52798484353	77510204157	78737478940	54660312521	24821999883	6659465121	801693126				329490080937
12	531441	1129201236	15337265481	94466866176	347808205150	850501243340	1450597853781	1761234683264	1522549279294	918577256532	368412718761	88417452312	9620848953			7428653405721
13	1594323	13010046984	191882322711	1292545286460	5257356981306	14383463040088	27890583804227	39312056531604	40590757037546	30475763577544	16228730899239	5818930847820	1261529321859	125072630712		182841683922423
14	4782969	164107554633	2605036185738	19018170732861	84559892654379	255527681050786	554291279270299	888128495445049	1064428030153487	954383577442010	632351131272431	300982333143909	97466423141502	19248453704493	1751021612937	4874905638147483

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