A122076 → D+1

nonnegative coefficients

Proved for all n: all roots real what this means

Recurrence

\(\left(12 + 10\,x - 8\,x^{2} - 12\,n - 26\,n x + 8\,n x^{2} - 8\,n^{2} + 16\,n^{2} x + 8\,n^{3}\right)\,P_{n}(x) = \left(66 + 58\,x - 26\,x^{2} - 24\,x^{3} - 90\,n - 138\,n x - 22\,n x^{2} + 24\,n x^{3} - 8\,n^{2} + 56\,n^{2} x + 48\,n^{2} x^{2} + 32\,n^{3} + 24\,n^{3} x\right)\,P_{n-1}(x) + \left(15 + 12\,x - 35\,x^{2} - 48\,x^{3} - 16\,x^{4} - 30\,n - 94\,n x - 96\,n x^{2} - 32\,n x^{3} - 20\,n^{2} - 36\,n^{2} x - 16\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 2\)

\(P_{1}(x) = 5 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	2															2
1	5	2														7
2	15	12	2													29
3	48	60	21	2												131
4	151	264	152	32	2											601
5	463	1050	865	310	45	2										2735
6	1390	3888	4206	2192	555	60	2									12293
7	4105	13678	18396	12712	4739	910	77	2								54619
8	11967	46320	74632	64544	32160	9184	1400	96	2							240305
9	34524	152316	286299	298074	186939	71838	16422	2052	117	2						1048583
10	98747	489300	1051780	1282240	972255	472628	146250	27600	2895	140	2					4543837
11	280415	1542222	3733103	5220006	4650195	2743510	1078693	277002	44154	3960	165	2				19573427
12	791450	4784832	12883116	20334416	20832762	14483760	6935348	2271984	495099	67848	5280	192	2			83886089
13	2222093	14649050	43435080	76407760	88554050	70976620	40160250	16072940	4484480	843700	100815	6890	221	2		357913951
14	6210351	44342004	143584910	278611760	360531360	327549040	214142460	101530080	34707855	8388380	1381380	145600	8827	252	2	1521134261

Computed in your browser ?

Technical details

Node id: n-04f443bf4d795a35 · revision 1 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-843b6dd431a207b6

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

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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:     2;
  1:     5,      2;
  2:    15,     12,      2;
  3:    48,     60,     21,      2;
  4:   151,    264,    152,     32,      2;
  5:   463,   1050,    865,    310,     45,     2;
  6:  1390,   3888,   4206,   2192,    555,    60,     2;
  7:  4105,  13678,  18396,  12712,   4739,   910,    77,    2;
  8: 11967,  46320,  74632,  64544,  32160,  9184,  1400,   96,   2;
  9: 34524, 152316, 286299, 298074, 186939, 71838, 16422, 2052, 117, 2;

%F Formula

The row polynomials satisfy (8*n^3-8*n^2-12*n+12+(16*n^2-26*n+10)*x+(8*n-8)*x^2)*P(n,x) = (32*n^3-8*n^2-90*n+66+(24*n^3+56*n^2-138*n+58)*x+(48*n^2-22*n-26)*x^2+(24*n-24)*x^3)*P(n-1,x) - (20*n^2+30*n-15+(36*n^2+94*n-12)*x+(16*n^2+96*n+35)*x^2+(32*n+48)*x^3+16*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 2, P(1,x) = 5 + 2*x.

%t Mathematica

P[0, x_] := 2;
P[1, x_] := 5 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((32*n^3 - 8*n^2 - 90*n + 66 + (24*n^3 + 56*n^2 - 138*n + 58)*x + (48*n^2 - 22*n - 26)*x^2 + (24*n - 24)*x^3)*P[n - 1, x] - (20*n^2 + 30*n - 15 + (36*n^2 + 94*n - 12)*x + (16*n^2 + 96*n + 35)*x^2 + (32*n + 48)*x^3 + 16*x^4)*D[P[n - 1, x], x])/(8*n^3 - 8*n^2 - 12*n + 12 + (16*n^2 - 26*n + 10)*x + (8*n - 8)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 5 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((32*n^3 - 8*n^2 - 90*n + 66 + (24*n^3 + 56*n^2 - 138*n + 58)*x + (48*n^2 - 22*n - 26)*x^2 + (24*n - 24)*x^3)*v[i - 1] - (20*n^2 + 30*n - 15 + (36*n^2 + 94*n - 12)*x + (16*n^2 + 96*n + 35)*x^2 + (32*n + 48)*x^3 + 16*x^4)*deriv(v[i - 1], x))/(8*n^3 - 8*n^2 - 12*n + 12 + (16*n^2 - 26*n + 10)*x + (8*n - 8)*x^2)); 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)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 0, 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-58ea4b3a2d7c4054

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 derivative_polynomial, shift 0 e-58ea4b3a2d7c4054

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): Coefficients of a generalized Jaco-Lucas polynomial (even indices) read by rows.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check three_term_recurrence e-644899f192503d96 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.
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023