A327997 → x ↦ x+1 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

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

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

\(P_{1}(x) = 4 + 5\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	4	5														9
2	17	43	27													87
3	78	299	379	159												915
4	393	2024	3875	3272	1029											10593
5	2208	14269	36556	46459	29316	7353										136161
6	13977	108351	346954	588001	556714	279387	58095									1951479
7	100026	900927	3449965	7287384	9176931	6893732	2861811	506691								31177467
8	806769	8248104	36630446	92365344	144721779	144357088	89559462	31608216	4860297							553157505
9	7280604	83041749	418346424	1222462142	2284570884	2832828711	2331526744	1228594518	376225740	51023277						10835900793
10	72865089	915206859	5144885577	17054250602	36929652829	54604283889	55848573367	39025828994	17835080715	4814585091	583097859					232828310871
11	801693126	10977783651	68005185183	251666889913	618393650876	1059671788251	1292491633973	1122355904911	680130319320	273968323749	66033977571	7215769575				5451712920099
12	9620848953	142482922560	963132891489	3930531767328	10788534694618	20987286512864	29676558920469	30739405149872	23152383970066	12367760973360	4448440334601	967414418064	96210083853			138269763488097
13	125072630712	1990359076653	14564887905348	64910185851081	196605220986604	427469138737354	686520574437556	824719589446037	741184015596740	492237647842082	234839347902732	76225042632465	15088647846780	1375803720801		3777855534612945
14	1751021612937	29780151457743	234378493933752	1131669203322867	3745775588381969	8992568717186198	16150147263719039	22045490789815351	22987078500752365	18222286865585518	10811429071699827	4655883800636667	1375858089850932	249750775650627	21012273704151	110654860607309943

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

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 basis_map_A007318 of Triangle read by rows: coefficients of the polynomials given by KummerU(-n, 1 - n - x, 3).?
  - Geometric basis map (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
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows: coefficients of the polynomials given by KummerU(-n, 1 - n - x, 3).?
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
  - 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 (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)

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