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	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	x¹⁷	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
9	236196	27500997	733164318	7013849751	29130163344	57619176930	56432679156	27349109982	6298587324	627010281	22213374	177147							185253868800
10	708588	113251683	4017987879	50769569613	280236727533	749647232382	1022672390502	722344716570	259901467302	45204146607	3418478595	91274505	531441						3138418483200
11	2125764	463458405	21653308632	356229596496	2564704841472	9053259515073	16646698975680	16373673817560	8611151526036	2355572943783	315043437768	18348081720	372892488	1594323					56317176115200
12	6377292	1887314403	115127321877	2437127575554	22539336161754	102948819837561	248710323077199	329234880404556	241389908774712	96902373152997	20539992814467	2143737446418	97242698562	1516547679	4782969				1067062284288000
13	19131876	7656077253	605461278786	16331418297117	191583492076536	1114290938342505	3465421308600174	6012484506446913	5931959949119556	3326980219663023	1039927901351262	173480626837623	14302808172576	510105894627	6145906866	14348907			21287892571545600
14	57395628	30963899811	3155944753875	107579019905763	1583930238530451	11576268050110575	45579982593161919	101476868296041855	131075331982274439	99010329655522473	43367293311841401	10727923640835849	1426781029989945	93881673011973	2653494936501	24837124821	43046721		446032034832384000
15	172186884	124931767269	16330932842076	698616589309182	12794570050370304	116197742832465798	571267896484355028	1604076067834552686	2653852143880030896	2624253680193798312	1550645720772192132	539476871222624202	107053749430160400	11475368040286122	607890861051084	13708401805386	100152038076	129140163	9792430582910976000

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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.