A108299 → x ↦ x+1

Proved for all n: all roots real what this means

Recurrence

\(\left(-3 + 2\,n\right)\,P_{n}(x) = \left(4 + 15\,x + 8\,x^{2} - 2\,n - 8\,n x - 4\,n x^{2}\right)\,P_{n-1}(x) + \left(3 + 11\,x + 12\,x^{2} + 4\,x^{3}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	0	-1														-1
3	-1	-3	-1													-5
4	-1	-2	1	1												-1
5	0	3	9	6	1											19
6	1	7	13	5	-2	-1										23
7	1	4	-2	-22	-24	-9	-1									-53
8	0	-5	-30	-60	-45	-9	3	1								-145
9	-1	-11	-37	-38	25	70	46	12	1							67
10	-1	-6	3	87	220	229	106	14	-4	-1						647
11	0	7	63	210	308	147	-105	-160	-75	-15	-1					379
12	1	15	73	123	-89	-609	-889	-615	-205	-20	5	1				-2209
13	1	8	-4	-220	-880	-1582	-1386	-392	295	305	111	18	1			-3725
14	0	-9	-108	-504	-1110	-918	810	2610	2619	1350	351	27	-6	-1		5111
15	-1	-19	-121	-284	214	2644	6240	7356	4494	847	-665	-518	-154	-21	-1	20011

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Technical details

Node id: n-e15113e58c389f73 · revision 1 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-830e017e05135d66

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 -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:  1;
   2:  0,  -1;
   3: -1,  -3,  -1;
   4: -1,  -2,   1,   1;
   5:  0,   3,   9,   6,   1;
   6:  1,   7,  13,   5,  -2,  -1;
   7:  1,   4,  -2, -22, -24,  -9,  -1;
   8:  0,  -5, -30, -60, -45,  -9,   3,  1;
   9: -1, -11, -37, -38,  25,  70,  46, 12,  1;
  10: -1,  -6,   3,  87, 220, 229, 106, 14, -4, -1;

%F Formula

The row polynomials satisfy (2*n-3)*P(n,x) = -(2*n-4+(8*n-15)*x+(4*n-8)*x^2)*P(n-1,x) + (3+11*x+12*x^2+4*x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 4 + (8*n - 15)*x + (4*n - 8)*x^2)*P[n - 1, x] + (3 + 11*x + 12*x^2 + 4*x^3)*D[P[n - 1, x], x])/(2*n - 3)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(2*n - 4 + (8*n - 15)*x + (4*n - 8)*x^2)*v[i - 1] + (3 + 11*x + 12*x^2 + 4*x^3)*deriv(v[i - 1], x))/(2*n - 3)); 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-b80b7af82ad2cf71

Proof chains

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

preserver_image affine, shift 0 e-b80b7af82ad2cf71

Affine, reversal and Möbius substitutions (elementary) polytool ?
- premise (sequence): Triangle read by rows, 0 <= k <= n: T(n,k) = binomial(n-[(k+1)/2],[k/2])*(-1)^[(k+1)/2].
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check three_term_recurrence e-7467667c9a732a94 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