A028338 → x ↦ (−2/9)x+1 (shift +2)

Proved for all n: all roots real, in [9/2, ∞) · alternating coefficient signs · interlacing (weak) what this means

Recurrence from OEIS A051231

\(P_{n}(x) = \left(-18 + x - 9\,n\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 162 - 27\,x + x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	162	-27	1								136
1	-4374	891	-54	1							-3536
2	157464	-36450	2835	-90	1						123760
3	-7085880	1797714	-164025	6885	-135	1					-5445440
4	382637520	-104162436	10655064	-535815	14175	-189	1				288608320
5	-24106163760	6944870988	-775431468	44411409	-1428840	26082	-252	1			-17893715840
6	1735643790720	-524136874896	62775936684	-3973052916	147287889	-3306744	44226	-324	1		1270453824640
7	-140587147048320	44190730657296	-5608987746300	384593222880	-15903371925	415134153	-6889050	70470	-405	1	-101636305971200

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-59f3eb88935e03d3 · revision 1 · JSON
Definition source: oeis_catalog:A051231
Fingerprint: f6-309bfae78f1d8828

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 2)
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:                  162,                -27,                  1;
  1:                -4374,                891,                -54,                1;
  2:               157464,             -36450,               2835,              -90,                1;
  3:             -7085880,            1797714,            -164025,             6885,             -135,             1;
  4:            382637520,         -104162436,           10655064,          -535815,            14175,          -189,             1;
  5:         -24106163760,         6944870988,         -775431468,         44411409,         -1428840,         26082,          -252,          1;
  6:        1735643790720,      -524136874896,        62775936684,      -3973052916,        147287889,      -3306744,         44226,       -324,         1;
  7:     -140587147048320,     44190730657296,     -5608987746300,     384593222880,     -15903371925,     415134153,      -6889050,      70470,      -405,      1;
  8:    12652843234348800,  -4117752906204960,    548999627824296,  -40222377805500,    1815896696130,  -53265445695,    1035148653,  -13231350,    106920,   -495,    1;
  9: -1252631480200531200, 420310380948639840, -58468716060810264, 4531015030568796, -219996150722370, 7089175819935, -155745162342, 2345052303, -23816430, 155925, -594, 1;

%F Formula

The row polynomials satisfy P(n,x) = -(9*n+18-x)*P(n-1,x) for n >= 1, with P(0,x) = 162 - 27*x + x^2.

%t Mathematica

P[0, x_] := 162 - 27*x + x^2;
P[n_, x_] := P[n, x] = Expand[-(9*n + 18 - 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] = 162 - 27*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = -(9*n + 18 - 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 [9/2, +inf)
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-2bc76bcaa23d62fb
Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n(-x) have one sign
All coefficients of each polynomial have the same sign (after x -> -x when alternating).

derived from the cited steps by a stated theorem: e-2bc76bcaa23d62fb
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-2bc76bcaa23d62fb

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 2 e-2bc76bcaa23d62fb

Affine, reversal and Möbius substitutions (elementary) polytool ?
- premise (sequence): Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.
  uses: Every row P_n, n >= 2, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-59b849691f316c6f root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - parent_provers root_domain e-ef36674c01230a86 root: theorem check
    
    Parent-row provers of polytool polytool ?

Used in 2 proof steps

e-07f8adddfb9eb09c obreschkoff_combination → obreschkoff_combination of Triangle read by rows: T(n, m) = S1(n, m)*9^(n-m), where S1 ar…
e-07f8adddfb9eb09c obreschkoff_combination → obreschkoff_combination of Triangle read by rows: T(n, m) = S1(n, m)*9^(n-m), where S1 ar…

Related sequences ?

No candidate equalities or related sequences.

References

[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003