A137338 → x ↦ x+1

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	0	1							1
2	-2	-1	1						-2
3	4	-3	-3	1					-1
4	-4	17	2	-6	1				10
5	-4	-57	24	21	-10	1			-25
6	44	179	-189	-45	65	-15	1		40
7	-236	-631	1145	-66	-365	148	-21	1	-25

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-b74f038f49f19922 · revision 1 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-9b5c4299a56ccb0f

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:    -2,     -1,     1;
  3:     4,     -3,    -3,      1;
  4:    -4,     17,     2,     -6,      1;
  5:    -4,    -57,    24,     21,    -10,     1;
  6:    44,    179,  -189,    -45,     65,   -15,     1;
  7:  -236,   -631,  1145,    -66,   -365,   148,   -21,   1;
  8:  1300,   2749, -7134,   1967,   1969, -1281,   287, -28,   1;
  9: -8276, -15013, 49516, -22276, -10500, 10885, -3388, 502, -36, 1;

%F Formula

The row polynomials satisfy P(n,x) = -n*P(n-2,x) - (n-1-x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := x;
P[n_, x_] := P[n, x] = Expand[-n*P[n - 2, x] - (n - 1 - 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; for(n = 2, N, my(i = n + 1); v[i] = -n*v[i - 2] - (n - 1 - 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-5aedd40a059ef120
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-5aedd40a059ef120

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 affine, shift 0 e-5aedd40a059ef120

Affine, reversal and Möbius substitutions (elementary) 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.
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023