A053123 → D+1

Proved for all n: all roots real what this means

Recurrence

\(\left(-30 + 38\,x - 8\,x^{2} + 62\,n - 54\,n x + 8\,n x^{2} - 40\,n^{2} + 16\,n^{2} x + 8\,n^{3}\right)\,P_{n}(x) = \left(-18 + 102\,x - 128\,x^{2} + 32\,x^{3} + 42\,n - 206\,n x + 162\,n x^{2} - 24\,n x^{3} - 32\,n^{2} + 128\,n^{2} x - 48\,n^{2} x^{2} + 8\,n^{3} - 24\,n^{3} x\right)\,P_{n-1}(x) + \left(-6\,x + 35\,x^{2} - 48\,x^{3} + 16\,x^{4} + 10\,n x - 48\,n x^{2} + 32\,n x^{3} - 4\,n^{2} x + 16\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	-1	-2														-3
3	-3	2	3													2
4	-5	14	-2	-4												3
5	-7	34	-39	0	5											-7
6	-9	62	-132	84	5	-6										4
7	-11	98	-305	384	-154	-14	7									5
8	-13	142	-582	1100	-930	252	28	-8								-11
9	-15	194	-987	2496	-3245	1980	-378	-48	9							6
10	-17	254	-1544	4900	-8645	8294	-3828	528	75	-10						7
11	-19	322	-2277	8704	-19460	25662	-19019	6864	-693	-110	11					-15
12	-21	398	-3210	14364	-38964	65688	-67704	40040	-11583	858	154	-12				8
13	-23	482	-4367	22400	-71535	147288	-195636	162656	-78650	18590	-1001	-208	13			9
14	-25	574	-5772	33396	-122815	299250	-488376	527136	-361998	145860	-28600	1092	273	-14		-19
15	-27	674	-7449	48000	-199870	563178	-1092861	1457376	-1308150	755820	-257686	42432	-1092	-350	15	10

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Technical details

Node id: n-0de906cd6e92b6be · revision 1 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-8ba428bd17294bc6

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:  -1,  -2;
   3:  -3,   2,     3;
   4:  -5,  14,    -2,   -4;
   5:  -7,  34,   -39,    0,     5;
   6:  -9,  62,  -132,   84,     5,   -6;
   7: -11,  98,  -305,  384,  -154,  -14,     7;
   8: -13, 142,  -582, 1100,  -930,  252,    28,  -8;
   9: -15, 194,  -987, 2496, -3245, 1980,  -378, -48,  9;
  10: -17, 254, -1544, 4900, -8645, 8294, -3828, 528, 75, -10;

%F Formula

The row polynomials satisfy (8*n^3-40*n^2+62*n-30+(16*n^2-54*n+38)*x+(8*n-8)*x^2)*P(n,x) = (8*n^3-32*n^2+42*n-18-(24*n^3-128*n^2+206*n-102)*x-(48*n^2-162*n+128)*x^2-(24*n-32)*x^3)*P(n-1,x) - ((4*n^2-10*n+6)*x-(16*n^2-48*n+35)*x^2-(32*n-48)*x^3-16*x^4)*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[((8*n^3 - 32*n^2 + 42*n - 18 - (24*n^3 - 128*n^2 + 206*n - 102)*x - (48*n^2 - 162*n + 128)*x^2 - (24*n - 32)*x^3)*P[n - 1, x] - ((4*n^2 - 10*n + 6)*x - (16*n^2 - 48*n + 35)*x^2 - (32*n - 48)*x^3 - 16*x^4)*D[P[n - 1, x], x])/(8*n^3 - 40*n^2 + 62*n - 30 + (16*n^2 - 54*n + 38)*x + (8*n - 8)*x^2)]];
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] = ((8*n^3 - 32*n^2 + 42*n - 18 - (24*n^3 - 128*n^2 + 206*n - 102)*x - (48*n^2 - 162*n + 128)*x^2 - (24*n - 32)*x^3)*v[i - 1] - ((4*n^2 - 10*n + 6)*x - (16*n^2 - 48*n + 35)*x^2 - (32*n - 48)*x^3 - 16*x^4)*deriv(v[i - 1], x))/(8*n^3 - 40*n^2 + 62*n - 30 + (16*n^2 - 54*n + 38)*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, 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-d32ce5c030354d3d

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-d32ce5c030354d3d

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): Triangle of coefficients of shifted Chebyshev's S(n,x-2) = U(n,x/2-1) polynomials (exponents of x in decreasing order).
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check three_term_recurrence e-b3aa6b8d6262c436 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