A185410 → Pₙ + Pₙ₊₁

nonnegative coefficients

Proved for all n: all roots real what this means

Recurrence

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

\(P_{0}(x) = 2\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	2								2
1	2	2							4
2	2	12	4						18
3	2	46	64	8					120
4	2	152	576	304	16				1050
5	2	474	4024	5464	1344	32			11340
6	2	1444	24628	69408	44288	5696	64		145530
7	2	4358	139752	725120	943520	325728	23552	128	2162160

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-c9ad85130c4da295 · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-c25478bcb8d85797

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): 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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0: 2;
  1: 2,     2;
  2: 2,    12,       4;
  3: 2,    46,      64,        8;
  4: 2,   152,     576,      304,        16;
  5: 2,   474,    4024,     5464,      1344,        32;
  6: 2,  1444,   24628,    69408,     44288,      5696,        64;
  7: 2,  4358,  139752,   725120,    943520,    325728,     23552,      128;
  8: 2, 13104,  757600,  6710624,  15614400,  11045248,   2249216,    96000,    256;
  9: 2, 39346, 3991120, 57343504, 219735328, 275840608, 116961664, 14888320, 388096, 512;

%F Formula

The row polynomials satisfy P(n,x) = (1+(2*n-4)*x)*P(n-2,x) + (2*x-2*x^2)*d/dx P(n-2,x) + 2*n*x*P(n-1,x) + (2*x-2*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 2, P(1,x) = 2 + 2*x.

%t Mathematica

P[0, x_] := 2;
P[1, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[(1 + (2*n - 4)*x)*P[n - 2, x] + (2*x - 2*x^2)*D[P[n - 2, x], x] + 2*n*x*P[n - 1, x] + (2*x - 2*x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 2 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (1 + (2*n - 4)*x)*v[i - 2] + (2*x - 2*x^2)*deriv(v[i - 2], x) + 2*n*x*v[i - 1] + (2*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 - 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-261fc200d1c6c7f8

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-261fc200d1c6c7f8

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): A decomposition of the double factorials A001147.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check liu_wang e-e2b66c2a000699df root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
- premise (sequence): A decomposition of the double factorials A001147.
  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.