A113953 → x ↦ x−5

Proved for all n: all roots real, in (−3, 5] · interlacing (weak) what this means

Recurrence

\(P_{n}(x) = \left(-5 + x\right)\,P_{n-1}(x) + \left(-10 + 2\,x\right)\,P_{n-2}(x)\)

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

\(P_{1}(x) = -5 + x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	-5	1														-4
2	15	-8	1													8
3	-25	35	-11	1												0
4	-25	-90	64	-14	1											-64
5	375	25	-230	102	-17	1										256
6	-1625	1100	355	-472	149	-20	1									-512
7	4375	-6625	1675	1235	-843	205	-23	1								0
8	-5625	23250	-16350	930	3016	-1370	270	-26	1							4096
9	-15625	-46875	75000	-30000	-3250	6130	-2080	344	-29	1						-16384
10	134375	-25000	-211875	183000	-42050	-14168	11090	-3000	427	-32	1					32768
11	-515625	696875	190625	-676875	365750	-39010	-36558	18490	-4157	519	-35	1				0
12	1234375	-3481250	1812500	1321250	-1719125	618380	4544	-76828	29005	-5578	620	-38	1			-262144
13	-1015625	10640625	-13056250	2356250	4905625	-3689425	883220	130668	-143303	43391	-7290	730	-41	1		1048576
14	-7265625	-16937500	50834375	-34425000	-2338125	13730700	-6914205	1007248	403477	-246468	62485	-9320	849	-44	1	-2097152

Generated b-files: A143462 (row sums) ?

Computed in your browser ?

Technical details

Node id: n-54865b1c531f7155 · revision 1 · JSON
Definition source: derived:basis_map_coefficients_of:oeis_catalog:A113953
Fingerprint: f6-074a9ec3b69568bd

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 ?

A143462: row sums, 15 terms agree (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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:     -5,      1;
  2:     15,     -8,      1;
  3:    -25,     35,    -11,      1;
  4:    -25,    -90,     64,    -14,     1;
  5:    375,     25,   -230,    102,   -17,     1;
  6:  -1625,   1100,    355,   -472,   149,   -20,     1;
  7:   4375,  -6625,   1675,   1235,  -843,   205,   -23,   1;
  8:  -5625,  23250, -16350,    930,  3016, -1370,   270, -26,   1;
  9: -15625, -46875,  75000, -30000, -3250,  6130, -2080, 344, -29, 1;

%F Formula

The row polynomials satisfy P(n,x) = -(5-x)*P(n-1,x) - (10-2*x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = -5 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := -5 + x;
P[n_, x_] := P[n, x] = Expand[-(5 - x)*P[n - 1, x] - (10 - 2*x)*P[n - 2, 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] = -5 + x; for(n = 2, N, my(i = n + 1); v[i] = -(5 - x)*v[i - 1] - (10 - 2*x)*v[i - 2]); 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 (-3, 5]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-0c61b86c2e25257b
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-0c61b86c2e25257b

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-0c61b86c2e25257b

Affine, reversal and Möbius substitutions (elementary) polytool ?
- premise (sequence): A Jacobsthal triangle.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-8, 0]
  - theorem_check ma_wang_interval e-2d5b16bf2955a407 root: theorem check
    
    Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ?

Used in 1 proof step

e-7b55c4767302dd2a basis_map (A038243) → A Jacobsthal triangle.

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