A137338 → D+1

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

Recurrence

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

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

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

\(P_{2}(x) = -3 - x + x^{2}\)

\(P_{3}(x) = 9 - 6\,x - 3\,x^{2} + x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	0	1														1
2	-3	-1	1													-3
3	9	-6	-3	1												1
4	-21	43	-4	-6	1											13
5	48	-215	104	11	-10	1										-61
6	-117	1083	-924	145	50	-15	1									223
7	285	-6112	7324	-2491	20	127	-21	1								-867
8	-273	39731	-59975	29421	-4506	-581	259	-28	1							4049
9	-7092	-296133	530100	-327840	83524	-4067	-2212	466	-36	1						-23289
10	126867	2496619	-5115295	3730294	-1253981	173901	7469	-5766	771	-45	1					160835
11	-1779855	-23489282	53887475	-44565291	18043645	-3660616	234801	49875	-12585	1200	-55	1				-1290687
12	24336099	243852003	-617360034	564738128	-262390254	66762753	-8324380	-10032	162318	-24585	1782	-66	1			11743733
13	-341686128	-2767588219	7654525744	-7613903751	3941002786	-1175147834	198459372	-13821835	-1335378	415338	-44396	2549	-78	1		-119121829
14	5014311675	34077523499	-102215153570	109203434851	-61733102733	20770319184	-4251549905	479672875	-10072645	-5546112	927487	-75517	3536	-91	1	1330692535

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Technical details

Node id: n-31240f0abddd4277 · revision 1 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-a6b7716422593ffa

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:     0,       1;
  2:    -3,      -1,      1;
  3:     9,      -6,     -3,       1;
  4:   -21,      43,     -4,      -6,     1;
  5:    48,    -215,    104,      11,   -10,     1;
  6:  -117,    1083,   -924,     145,    50,   -15,     1;
  7:   285,   -6112,   7324,   -2491,    20,   127,   -21,   1;
  8:  -273,   39731, -59975,   29421, -4506,  -581,   259, -28,   1;
  9: -7092, -296133, 530100, -327840, 83524, -4067, -2212, 466, -36, 1;

%F Formula

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

%t Mathematica

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

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = x; v[3] = -3 - x + x^2; v[4] = 9 - 6*x - 3*x^2 + x^3; for(n = 4, N, my(i = n + 1); v[i] = -(n^2 - 3*n + 2)*v[i - 4] - (2*n^2 - 5*n + 3 - (2*n - 2)*x)*v[i - 3] - (n^2 - (2*n - 2)*x + x^2)*v[i - 2] - (2*n - 1 - 2*x)*v[i - 1]); 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-6aed627db0c42e8b
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-6aed627db0c42e8b

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-6aed627db0c42e8b

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): Triangle read by rows: T(n,k), 0 <= k <= n, gives the coefficients of the Charlier polynomials (with parameter a=1), or…
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check three_term_recurrence e-fe1ae6008805b5f0 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