A257609 → Pₙ + Pₙ₊₁

nonnegative coefficients

Proved for all n: all roots real what this means

Recurrence

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

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

\(P_{1}(x) = 6 + 18\,x + 4\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	P_n(1)
0	3	2								5
1	6	18	4							28
2	12	104	92	8						216
3	24	504	1144	424	16					2112
4	48	2240	10720	10080	1840	32				24960
5	96	9504	85888	164288	78048	7712	64			345600
6	192	39296	625728	2153856	2075456	557184	31680	128		5483520
7	384	160128	4289152	24587136	41983872	23137408	3771264	128640	256	98058240

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-4b2633a9a6081341 · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-6d9e12a32349f2fa

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): 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:    3,       2;
  1:    6,      18,         4;
  2:   12,     104,        92,          8;
  3:   24,     504,      1144,        424,          16;
  4:   48,    2240,     10720,      10080,        1840,          32;
  5:   96,    9504,     85888,     164288,       78048,        7712,          64;
  6:  192,   39296,    625728,    2153856,     2075456,      557184,       31680,        128;
  7:  384,  160128,   4289152,   24587136,    41983872,    23137408,     3771264,     128640,       256;
  8:  768,  647168,  28233728,  255646208,   710885888,   693489152,   236797952,   24622592,    518912,     512;
  9: 1536, 2603520, 180794368, 2489430016, 10642729984, 16772531200, 10204887040, 2280865792, 156818944, 2085376, 1024;

%F Formula

The row polynomials satisfy P(n,x) = (2+(2*n-2)*x)*P(n-2,x) + (2*x-2*x^2)*d/dx P(n-2,x) + (1+(2*n+2)*x)*P(n-1,x) + (2*x-2*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 3 + 2*x, P(1,x) = 6 + 18*x + 4*x^2.

%t Mathematica

P[0, x_] := 3 + 2*x;
P[1, x_] := 6 + 18*x + 4*x^2;
P[n_, x_] := P[n, x] = Expand[(2 + (2*n - 2)*x)*P[n - 2, x] + (2*x - 2*x^2)*D[P[n - 2, x], x] + (1 + (2*n + 2)*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] = 3 + 2*x; v[2] = 6 + 18*x + 4*x^2; for(n = 2, N, my(i = n + 1); v[i] = (2 + (2*n - 2)*x)*v[i - 2] + (2*x - 2*x^2)*deriv(v[i - 2], x) + (1 + (2*n + 2)*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-1d58a9b88feb4822

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-1d58a9b88feb4822

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 2.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check liu_wang e-7ce0c29875c331b6 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - parent_provers root_domain e-f2db3ed118a27f94 root: theorem check
    
    Parent-row provers of polytool polytool ?
- premise (sequence): Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 2.
  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.