basis-map coefficients of A371080, shifted → x ↦ x−1

only rational roots

Proved for all n: all roots real, in (−∞, 1] what this means

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	-1	1							0
2	-2	1	1						0
3	-10	3	6	1					0
4	-80	14	51	14	1				0
5	-880	74	575	205	25	1			0
6	-12320	156	8124	3445	555	39	1		0
7	-209440	-9668	138264	66689	12880	1218	56	1	0

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-2e038ac8288fcb6a · revision 1 · JSON
Definition source: derived:basis_map_coefficients_of:oeis_catalog:A371080
Fingerprint: f6-1f79d2d40deced4b

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): yes

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:        -1,         1;
  2:        -2,         1,        1;
  3:       -10,         3,        6,        1;
  4:       -80,        14,       51,       14,       1;
  5:      -880,        74,      575,      205,      25,       1;
  6:    -12320,       156,     8124,     3445,     555,      39,     1;
  7:   -209440,     -9668,   138264,    66689,   12880,    1218,    56,    1;
  8:  -4188800,   -402800,  2755612,  1472044,  324289,   37240,  2338,   76,  1;
  9: -96342400, -13453200, 62976276, 36612624, 8930691, 1180809, 91014, 4086, 99, 1;

%F Formula

The row polynomials satisfy P(n,x) = (3*n-4+x)*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[(3*n - 4 + 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; for(n = 1, N, my(i = n + 1); v[i] = (3*n - 4 + 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, 1]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-c5f7781ad297615f

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-c5f7781ad297615f

Affine, reversal and Möbius substitutions (elementary) polytool ?
- premise (sequence): basis-map coefficients of Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {k : k mod m = 1 and 1…
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers root_domain e-aa6a1f197b91b3b6 root: theorem check
    
    Parent-row provers of polytool polytool ?

Used in 1 proof step

e-9c5e956579680e26 basis_map (A008277) → Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {k : k mod m = 1 a…

Related sequences ?

No candidate equalities or related sequences.