A257180 → Pₙ + Pₙ₊₁

nonnegative coefficients

Proved for all n: all roots real what this means

Recurrence

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

\(P_{1}(x) = 36 + 165\,x + 150\,x^{2} + 27\,x^{3}\)

\(P_{2}(x) = 108 + 867\,x + 1551\,x^{2} + 753\,x^{3} + 81\,x^{4}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	P_n(1)
1	36	165	150	27								378
2	108	867	1551	753	81							3360
3	324	4197	12816	12132	3552	243						33264
4	972	19299	92733	148278	84798	16071	729					362880
5	2916	85701	614586	1527393	1471848	549087	70602	2187				4324320
6	8748	371235	3828411	13999719	20857815	13142457	3365025	303549	6561			55883520
7	26244	1578981	22784532	117920754	256611936	249753840	108606708	19790742	1284180	19683		778377600
8	78732	6624291	130990089	932074692	2844961140	4031458914	2713128054	846252516	112778328	5366355	59049	11623772160

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-41f5536fe04f4d13 · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-3d34973da4d1ff05

Observed on the stored rows ?

Degrees: 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 (affine: slope 1, intercept 2)
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:     36,       165,        150,          27;
   2:    108,       867,       1551,         753,           81;
   3:    324,      4197,      12816,       12132,         3552,          243;
   4:    972,     19299,      92733,      148278,        84798,        16071,           729;
   5:   2916,     85701,     614586,     1527393,      1471848,       549087,         70602,         2187;
   6:   8748,    371235,    3828411,    13999719,     20857815,     13142457,       3365025,       303549,         6561;
   7:  26244,   1578981,   22784532,   117920754,    256611936,    249753840,     108606708,     19790742,      1284180,       19683;
   8:  78732,   6624291,  130990089,   932074692,   2844961140,   4031458914,    2713128054,    846252516,    112778328,     5366355,      59049;
   9: 236196,  27500997,  733164318,  7013849751,  29130163344,  57619176930,   56432679156,  27349109982,   6298587324,   627010281,   22213374,   177147;
  10: 708588, 113251683, 4017987879, 50769569613, 280236727533, 749647232382, 1022672390502, 722344716570, 259901467302, 45204146607, 3418478595, 91274505, 531441;

%F Formula

The row polynomials satisfy P(n,x) = (3+(n+2)*x)*P(n-2,x) + (x-x^2)*d/dx P(n-2,x) + (2+(n+4)*x)*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 36 + 165*x + 150*x^2 + 27*x^3, P(2,x) = 108 + 867*x + 1551*x^2 + 753*x^3 + 81*x^4.

%t Mathematica

P[1, x_] := 36 + 165*x + 150*x^2 + 27*x^3;
P[2, x_] := 108 + 867*x + 1551*x^2 + 753*x^3 + 81*x^4;
P[n_, x_] := P[n, x] = Expand[(3 + (n + 2)*x)*P[n - 2, x] + (x - x^2)*D[P[n - 2, x], x] + (2 + (n + 4)*x)*P[n - 1, x] + (x - 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] = 36 + 165*x + 150*x^2 + 27*x^3; v[2] = 108 + 867*x + 1551*x^2 + 753*x^3 + 81*x^4; for(n = 3, N, my(i = n); v[i] = (3 + (n + 2)*x)*v[i - 2] + (x - x^2)*deriv(v[i - 2], x) + (2 + (n + 4)*x)*v[i - 1] + (x - 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-52155003e339a993

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-52155003e339a993

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(x) = x + 3.
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check liu_wang e-27a325e06cca4d67 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) where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(x) = x + 3.
  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.