A356546 → Pₙ + Pₙ₊₁

nonnegative coefficients only rational roots

Proved for all n: all roots real what this means

Recurrence

\(\left(-2 - 2\,x + 3\,n + 2\,n x + 5\,n^{2} + 4\,n^{2} x\right)\,P_{n}(x) = \left(-6 - 10\,x - 4\,x^{2} + 2\,n + 2\,n x + 20\,n^{2} + 36\,n^{2} x + 16\,n^{2} x^{2}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 3 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	P_n(1)
0	3	2								5
1	8	14	6							28
2	26	72	66	20						184
3	90	340	480	300	70					1280
4	322	1540	2940	2800	1330	252				9184
5	1176	6804	16380	21000	15120	5796	924			67200
6	4356	29568	85932	138600	133980	77616	24948	3432		498432
7	16302	126984	432432	840840	1021020	792792	384384	106392	12870	3734016

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-fb0ded5b295bc8c5 · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-2d051ac784554c2b

Observed on the stored rows ?

Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (affine: slope 1, intercept 1)
Palindromic rows: no
γ-positive (gamma_positive): no
Nonnegative coefficients (nonn): yes
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:      3,       2;
  1:      8,      14,        6;
  2:     26,      72,       66,       20;
  3:     90,     340,      480,      300,       70;
  4:    322,    1540,     2940,     2800,     1330,      252;
  5:   1176,    6804,    16380,    21000,    15120,     5796,      924;
  6:   4356,   29568,    85932,   138600,   133980,    77616,    24948,     3432;
  7:  16302,  126984,   432432,   840840,  1021020,   792792,   384384,   106392,   12870;
  8:  61490,  540540,  2110680,  4804800,  7027020,  6846840,  4444440,  1853280,  450450,   48620;
  9: 233376, 2285140, 10064340, 26254800, 44924880, 52684632, 42882840, 23921040, 8751600, 1896180, 184756;

%F Formula

The row polynomials satisfy (5*n^2+3*n-2+(4*n^2+2*n-2)*x)*P(n,x) = (20*n^2+2*n-6+(36*n^2+2*n-10)*x+(16*n^2-4)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 3 + 2*x.

%t Mathematica

P[0, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((20*n^2 + 2*n - 6 + (36*n^2 + 2*n - 10)*x + (16*n^2 - 4)*x^2)*P[n - 1, x])/(5*n^2 + 3*n - 2 + (4*n^2 + 2*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] = 3 + 2*x; for(n = 1, N, my(i = n + 1); v[i] = ((20*n^2 + 2*n - 6 + (36*n^2 + 2*n - 10)*x + (16*n^2 - 4)*x^2)*v[i - 1])/(5*n^2 + 3*n - 2 + (4*n^2 + 2*n - 2)*x)); 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-9dcb6fc37a08fff7

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

obreschkoff_combination e-9dcb6fc37a08fff7

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check liu_wang e-883c9c594ff729de root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
- premise (sequence): Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).
  uses: Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
  
  no recorded property covers this requirement

Related sequences ?

No candidate equalities or related sequences.

References

[Fis06] Steve Fisk. Polynomials, roots, and interlacing. arXiv:math/0612833. arXiv
[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
[Obr63] Nikola Obreschkoff. Verteilung und berechnung der nullstellen reeller polynome. Hochschulbücher für mathematik, VEB Deutscher Verlag der Wissenschaften, 1963.