basis-map coefficients of A112227 → basis A112227

Proved for all n: all roots real what this means

Recurrence from OEIS A112227

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

\(P_{1}(x) = -2 + x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
1	-2	0	1															-1
2	0	-6	0	1														-5
3	12	0	-12	0	1													1
4	0	60	0	-20	0	1												41
5	-120	0	180	0	-30	0	1											31
6	0	-840	0	420	0	-42	0	1										-461
7	1680	0	-3360	0	840	0	-56	0	1									-895
8	0	15120	0	-10080	0	1512	0	-72	0	1								6481
9	-30240	0	75600	0	-25200	0	2520	0	-90	0	1							22591
10	0	-332640	0	277200	0	-55440	0	3960	0	-110	0	1						-107029
11	665280	0	-1995840	0	831600	0	-110880	0	5940	0	-132	0	1					-604031
12	0	8648640	0	-8648640	0	2162160	0	-205920	0	8580	0	-156	0	1				1964665
13	-17297280	0	60540480	0	-30270240	0	5045040	0	-360360	0	12012	0	-182	0	1			17669471
14	0	-259459200	0	302702400	0	-90810720	0	10810800	0	-600600	0	16380	0	-210	0	1		-37341149
15	518918400	0	-2075673600	0	1210809600	0	-242161920	0	21621600	0	-960960	0	21840	0	-240	0	1	-567425279

Computed in your browser ?

Technical details

Node id: n-0ce01d5a88917ba1 · revision 1 · JSON
Definition source: oeis_catalog:A112227
Fingerprint: f6-a0263e27252ad1fd

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (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 ?

A293604: row sums, 15 terms agree (row-sum match: finite agreement only) · view at the OEIS

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:     -2,       0,     1;
   2:      0,      -6,     0,      1;
   3:     12,       0,   -12,      0,      1;
   4:      0,      60,     0,    -20,      0,      1;
   5:   -120,       0,   180,      0,    -30,      0,    1;
   6:      0,    -840,     0,    420,      0,    -42,    0,    1;
   7:   1680,       0, -3360,      0,    840,      0,  -56,    0,   1;
   8:      0,   15120,     0, -10080,      0,   1512,    0,  -72,   0,    1;
   9: -30240,       0, 75600,      0, -25200,      0, 2520,    0, -90,    0, 1;
  10:      0, -332640,     0, 277200,      0, -55440,    0, 3960,   0, -110, 0, 1;

%F Formula

The row polynomials satisfy P(n,x) = x*P(n-1,x) - 2*d/dx P(n-1,x) for n >= 2, with P(1,x) = -2 + x^2.

%t Mathematica

P[1, x_] := -2 + x^2;
P[n_, x_] := P[n, x] = Expand[x*P[n - 1, x] - 2*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = -2 + x^2; for(n = 2, N, my(i = n); v[i] = x*v[i - 1] - 2*deriv(v[i - 1], x)); 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-77b4dff64dd0f71c

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

basis_map A112227 e-77b4dff64dd0f71c

Hermite-type basis map [MW08, Thm 2] polytool ?
- premise (sequence): basis-map coefficients of A scaled Hermite triangle.
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers periodic_product e-470643e70af29063 root: theorem check
    
    Parent-row provers of polytool polytool ?
  - preserver_image mobius, shift 0 e-f9cce4eb6656235e
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of A scaled Hermite triangle.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ parent_provers periodic_product e-470643e70af29063 (expanded above)

Related sequences ?

No candidate equalities or related sequences.

References

[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741