A267482 → D+1

Proved for all n: all roots real · interlacing (weak) what this means

Recurrence

\(\left(-6 - 2\,x + x^{2} + 13\,n + 4\,n x - 2\,n x^{2} - 3\,n^{2} + 2\,n^{3}\right)\,P_{n}(x) = \left(6 + 2\,x - x^{2} + 5\,n + 6\,n x + n x^{2} - n x^{3} + n^{2} + n^{2} x + n^{3} x\right)\,P_{n-1}(x) + \left(-24 + 10\,x^{2} - x^{4} - 4\,n^{2} + n^{2} x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	1														3
2	0	3	1													4
3	-3	0	4	1												2
4	-1	-8	0	5	1											-3
5	4	-3	-15	0	6	1										-7
6	2	15	-6	-24	0	7	1									-5
7	-5	8	36	-10	-35	0	8	1								3
8	-3	-24	20	70	-15	-48	0	9	1							10
9	6	-15	-70	40	120	-21	-63	0	10	1						8
10	4	35	-45	-160	70	189	-28	-80	0	11	1					-3
11	-7	24	120	-105	-315	112	280	-36	-99	0	12	1				-13
12	-5	-48	84	315	-210	-560	168	396	-45	-120	0	13	1			-11
13	8	-35	-189	224	700	-378	-924	240	540	-55	-143	0	14	1		3
14	6	63	-140	-560	504	1386	-630	-1440	330	715	-66	-168	0	15	1	16

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Technical details

Node id: n-c21445b3f1970578 · revision 1 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-d9a061640fcdc5b7

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): 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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:  1;
  1:  2,   1;
  2:  0,   3,   1;
  3: -3,   0,   4,   1;
  4: -1,  -8,   0,   5,   1;
  5:  4,  -3, -15,   0,   6,   1;
  6:  2,  15,  -6, -24,   0,   7,   1;
  7: -5,   8,  36, -10, -35,   0,   8, 1;
  8: -3, -24,  20,  70, -15, -48,   0, 9,  1;
  9:  6, -15, -70,  40, 120, -21, -63, 0, 10, 1;

%F Formula

The row polynomials satisfy (2*n^3-3*n^2+13*n-6+(4*n-2)*x-(2*n-1)*x^2)*P(n,x) = (n^2+5*n+6+(n^3+n^2+6*n+2)*x+(n-1)*x^2-n*x^3)*P(n-1,x) - (4*n^2+24-(n^2+10)*x^2+x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + 5*n + 6 + (n^3 + n^2 + 6*n + 2)*x + (n - 1)*x^2 - n*x^3)*P[n - 1, x] - (4*n^2 + 24 - (n^2 + 10)*x^2 + x^4)*D[P[n - 1, x], x])/(2*n^3 - 3*n^2 + 13*n - 6 + (4*n - 2)*x - (2*n - 1)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((n^2 + 5*n + 6 + (n^3 + n^2 + 6*n + 2)*x + (n - 1)*x^2 - n*x^3)*v[i - 1] - (4*n^2 + 24 - (n^2 + 10)*x^2 + x^4)*deriv(v[i - 1], x))/(2*n^3 - 3*n^2 + 13*n - 6 + (4*n - 2)*x - (2*n - 1)*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-50c0ba76f50a7e5c
Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
The roots of consecutive polynomials alternate on the real line.

by the cited steps: e-50c0ba76f50a7e5c

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-50c0ba76f50a7e5c

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): Triangle of coefficients of Gaussian polynomials [2n+1,1]_q represented as finite sum of terms (1+q^2)^k*q^(g-k), where…
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check three_term_recurrence e-d1dad2b285ccdcec root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
  - theorem_check three_term_recurrence e-d92e4a9b9719ac57 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