Triangle read by rows: coefficients of a Bessel polynomial recursion: P(x, n) = 2(n-1)P(x, n - 1)/x - n*P(x, n - 2) with substitution x -> 1/y.

Proved for all n: all roots real, in (−2, 2) · interlacing what this means

Recurrence from OEIS A136668

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

\(P_{1}(x) = -2 + 2\,x^{2}\)

\(P_{2}(x) = -11\,x + 8\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
1	-2	0	2															0
2	0	-11	0	8														-3
3	8	0	-74	0	48													-18
4	0	119	0	-632	0	384												-129
5	-48	0	1634	0	-6608	0	3840											-1182
6	0	-1409	0	24032	0	-81984	0	46080										-13281
7	384	0	-32798	0	389312	0	-1178496	0	645120									-176478
8	0	18825	0	-741056	0	6966848	0	-19270656	0	10321920								-2704119
9	-3840	0	666830	0	-17232128	0	137188224	0	-353323008	0	185794560							-46909362
10	0	-283875	0	21488216	0	-421277888	0	2955741696	0	-7180001280	0	3715891200						-908441931
11	46080	0	-14247210	0	679526288	0	-10914372224	0	69266193408	0	-160189562880	0	81749606400					-19422810138
12	0	4796295	0	-621279848	0	21785243456	0	-300369575424	0	1755728658432	0	-3892856094720	0	1961990553600				-454337698209
13	-645120	0	324164610	0	-25666644080	0	719217540992	0	-8779335668736	0	47891598999552	0	-102358752952320	0	51011754393600			-11540860811502
14	0	-90007785	0	18395806800	0	-1045444686080	0	24643634779136	0	-272157328601088	0	1399357613408256	0	-2895474940968960	0	1428329123020800		-316329037248921
15	10321920	0	-7886867310	0	962540509280	0	-42870821238272	0	879778414073856	0	-8930985442025472	0	43618468449484800	0	-87680436299366400	0	42849873690624000	-9305217344483598

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Technical details

Node id: n-1eae5c7270ad0739 · revision 1 · JSON
Definition source: oeis_catalog:A136668
Fingerprint: f6-f01cd712f19a05d6

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 1)
Palindromic rows: no
γ-positive (gamma_positive): no
Nonnegative coefficients (nonn): no
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

OEIS matches ?

No OEIS matches.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:    -2,       0,      2;
   2:     0,     -11,      0,        8;
   3:     8,       0,    -74,        0,        48;
   4:     0,     119,      0,     -632,         0,        384;
   5:   -48,       0,   1634,        0,     -6608,          0,      3840;
   6:     0,   -1409,      0,    24032,         0,     -81984,         0,      46080;
   7:   384,       0, -32798,        0,    389312,          0,  -1178496,          0,     645120;
   8:     0,   18825,      0,  -741056,         0,    6966848,         0,  -19270656,          0,    10321920;
   9: -3840,       0, 666830,        0, -17232128,          0, 137188224,          0, -353323008,           0, 185794560;
  10:     0, -283875,      0, 21488216,         0, -421277888,         0, 2955741696,          0, -7180001280,         0, 3715891200;

%F Formula

The row polynomials satisfy P(n,x) = 2*n*x*P(n-1,x) - (n+1)*P(n-2,x) for n >= 3, with P(1,x) = -2 + 2*x^2, P(2,x) = -11*x + 8*x^3.

%t Mathematica

P[1, x_] := -2 + 2*x^2;
P[2, x_] := -11*x + 8*x^3;
P[n_, x_] := P[n, x] = Expand[2*n*x*P[n - 1, x] - (n + 1)*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = -2 + 2*x^2; v[2] = -11*x + 8*x^3; for(n = 3, N, my(i = n); v[i] = 2*n*x*v[i - 1] - (n + 1)*v[i - 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)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-65533d28bec86ec5
Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (strict, previous interlaces next) for every m >= 2
The roots of consecutive polynomials alternate on the real line.

by the cited steps: e-65533d28bec86ec5, e-f1f9c50c96bd4da9
Real-rooted. Every row P_n, n >= 1, is zero or real-rooted with all roots in (-2, 2)
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-f1f9c50c96bd4da9

Proof chains

2 incoming proof steps. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

theorem_check three_term_recurrence e-65533d28bec86ec5 root: theorem check

Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
theorem_check three_term_recurrence e-f1f9c50c96bd4da9 root: theorem check

Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

Related sequences ?

No candidate equalities or related sequences.

References

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