A122076 → D

nonnegative coefficients

Proved for all n: all roots real what this means

Recurrence

\(\left(3 - 5\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(-14\,n - 11\,n x + 8\,n^{2} + 6\,n^{2} x\right)\,P_{n-1}(x) + \left(-5\,n - 9\,n x - 4\,n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 2\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	2														2
2	8	4													12
3	30	30	6												66
4	104	160	72	8											344
5	340	710	510	140	10										1710
6	1068	2820	2796	1260	240	12									8196
7	3262	10416	13188	8316	2660	378	14								38234
8	9760	36560	56352	45760	20720	5040	560	16							174768
9	28746	123570	224514	223236	131130	45612	8820	792	18						786438
10	83620	405680	848940	999260	722440	328140	91560	14520	1080	20					3495260
11	240812	1301410	3082398	4192540	3602060	2023098	741510	171072	22770	1430	22				15379122
12	687768	4097064	10834584	16722888	16652040	11153352	5076456	1546776	301752	34320	1848	24			67108872
13	1950650	12698400	37085880	64045800	72542600	56468100	30748900	11680240	3024450	507650	50050	2340	26		290805086
14	5499704	38842300	124163760	237223840	301225400	267303960	169591800	77302680	25045020	5605600	820820	70980	2912	28	1252698804

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Technical details

Node id: n-aa424fc3fffbd09d · revision 1 · JSON
Definition source: derived:derivative
Fingerprint: f6-5870c1c52d074a69

Observed on the stored rows ?

Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
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:     0;
  1:     2;
  2:     8,      4;
  3:    30,     30,      6;
  4:   104,    160,     72,      8;
  5:   340,    710,    510,    140,     10;
  6:  1068,   2820,   2796,   1260,    240,    12;
  7:  3262,  10416,  13188,   8316,   2660,   378,   14;
  8:  9760,  36560,  56352,  45760,  20720,  5040,  560,  16;
  9: 28746, 123570, 224514, 223236, 131130, 45612, 8820, 792, 18;

%F Formula

The row polynomials satisfy (2*n^2-5*n+3)*P(n,x) = (8*n^2-14*n+(6*n^2-11*n)*x)*P(n-1,x) - (5*n+9*n*x+4*n*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^2 - 14*n + (6*n^2 - 11*n)*x)*P[n - 1, x] - (5*n + 9*n*x + 4*n*x^2)*D[P[n - 1, x], x])/(2*n^2 - 5*n + 3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((8*n^2 - 14*n + (6*n^2 - 11*n)*x)*v[i - 1] - (5*n + 9*n*x + 4*n*x^2)*deriv(v[i - 1], x))/(2*n^2 - 5*n + 3)); 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-fa6dba774ff7b5f5

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-fa6dba774ff7b5f5

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 ?

Same fingerprint (3)

Equal up to content, sign, reversal and a small index shift.

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