A142075 → Pₙ + Pₙ₊₁

nonnegative coefficients

Proved for all n: all roots real what this means

Recurrence

\(P_{n}(x) = \left(1 - 4\,x + 2\,n x\right)\,P_{n-2}(x) + \left(x - 2\,x^{2}\right)\,P_{n-2}'(x) + \left(2\,n x\right)\,P_{n-1}(x) + \left(x - 2\,x^{2}\right)\,P_{n-1}'(x)\)

\(P_{1}(x) = 2 + 2\,x\)

\(P_{2}(x) = 2 + 10\,x + 4\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	P_n(1)
1	2	2								4
2	2	10	4							16
3	2	30	48	8						88
4	2	74	308	216	16					616
5	2	166	1472	2624	928	32				5224
6	2	354	5972	21744	19968	3872	64			51976
7	2	734	21936	144280	268960	141216	15872	128		593128
8	2	1498	75604	830824	2748944	2960864	950720	64384	256	7633096

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-ada7ab369cbe905b · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-2d7bbe3db765a185

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 0)
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): no

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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1: 2,    2;
   2: 2,   10,      4;
   3: 2,   30,     48,        8;
   4: 2,   74,    308,      216,        16;
   5: 2,  166,   1472,     2624,       928,        32;
   6: 2,  354,   5972,    21744,     19968,      3872,        64;
   7: 2,  734,  21936,   144280,    268960,    141216,     15872,       128;
   8: 2, 1498,  75604,   830824,   2748944,   2960864,    950720,     64384,      256;
   9: 2, 3030, 249792,  4347408,  23464704,  44754816,  30067200,   6187776,   259584,     512;
  10: 2, 6098, 801908, 21269440, 176775488, 545107264, 652371584, 288169984, 39334400, 1042944, 1024;

%F Formula

The row polynomials satisfy P(n,x) = (1+(2*n-4)*x)*P(n-2,x) + (x-2*x^2)*d/dx P(n-2,x) + 2*n*x*P(n-1,x) + (x-2*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 2 + 2*x, P(2,x) = 2 + 10*x + 4*x^2.

%t Mathematica

P[1, x_] := 2 + 2*x;
P[2, x_] := 2 + 10*x + 4*x^2;
P[n_, x_] := P[n, x] = Expand[(1 + (2*n - 4)*x)*P[n - 2, x] + (x - 2*x^2)*D[P[n - 2, x], x] + 2*n*x*P[n - 1, x] + (x - 2*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 + 2*x; v[2] = 2 + 10*x + 4*x^2; for(n = 3, N, my(i = n); v[i] = (1 + (2*n - 4)*x)*v[i - 2] + (x - 2*x^2)*deriv(v[i - 2], x) + 2*n*x*v[i - 1] + (x - 2*x^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-111ed7b99403026c

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-111ed7b99403026c

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): Triangle T(n, k) = 2^(k-1) * E(n, k-1) where E(n,k) are the Eulerian numbers A173018, read by rows.
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check liu_wang e-5f2e9a40e80d3367 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
- premise (sequence): Triangle T(n, k) = 2^(k-1) * E(n, k-1) where E(n,k) are the Eulerian numbers A173018, read by rows.
  uses: Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
  
  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.