A257613 → Pₙ + Pₙ₊₁

nonnegative coefficients

Proved for all n: all roots real what this means

Recurrence

\(P_{n}(x) = \left(4 + 2\,n x\right)\,P_{n-2}(x) + \left(2\,x - 2\,x^{2}\right)\,P_{n-2}'(x) + \left(3 + 4\,x + 2\,n x\right)\,P_{n-1}(x) + \left(2\,x - 2\,x^{2}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 5 + 4\,x\)

\(P_{1}(x) = 20 + 52\,x + 16\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	P_n(1)
0	5	4								9
1	20	52	16							88
2	80	464	432	64						1040
3	320	3552	7072	3200	256					14400
4	1280	25024	91264	87744	22144	1024				228480
5	5120	167552	1024128	1776768	961408	146688	4096			4085760
6	20480	1085184	10494976	29888000	29135360	9701120	943616	16384		81285120
7	81920	6871552	101031424	443930624	704896000	425290240	92415488	5948416	65536	1780531200

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-b018e3c7840cd649 · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-f0f0549965a63c8b

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:       5,         4;
  1:      20,        52,         16;
  2:      80,       464,        432,          64;
  3:     320,      3552,       7072,        3200,          256;
  4:    1280,     25024,      91264,       87744,        22144,         1024;
  5:    5120,    167552,    1024128,     1776768,       961408,       146688,         4096;
  6:   20480,   1085184,   10494976,    29888000,     29135360,      9701120,       943616,       16384;
  7:   81920,   6871552,  101031424,   443930624,    704896000,    425290240,     92415488,     5948416,      65536;
  8:  327680,  42834944,  930060288,  6036698112,  14617389056,  14356423680,   5712439296,   844516352,   36968448,    262144;
  9: 1310720, 264087552, 8285317120, 76925114368, 271164368896, 404279977984, 262844643328, 72142735360, 7483635712, 227446784, 1048576;

%F Formula

The row polynomials satisfy P(n,x) = (4+2*n*x)*P(n-2,x) + (2*x-2*x^2)*d/dx P(n-2,x) + (3+(2*n+4)*x)*P(n-1,x) + (2*x-2*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 5 + 4*x, P(1,x) = 20 + 52*x + 16*x^2.

%t Mathematica

P[0, x_] := 5 + 4*x;
P[1, x_] := 20 + 52*x + 16*x^2;
P[n_, x_] := P[n, x] = Expand[(4 + 2*n*x)*P[n - 2, x] + (2*x - 2*x^2)*D[P[n - 2, x], x] + (3 + (2*n + 4)*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] = 5 + 4*x; v[2] = 20 + 52*x + 16*x^2; for(n = 2, N, my(i = n + 1); v[i] = (4 + 2*n*x)*v[i - 2] + (2*x - 2*x^2)*deriv(v[i - 2], x) + (3 + (2*n + 4)*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-2b0de764d6dd6b90

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-2b0de764d6dd6b90

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 + 4.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check liu_wang e-5e8558eb5869f0da root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
- 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 + 4.
  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.