A290315 → Pₙ + Pₙ₊₁

nonnegative coefficients

Proved for all n: all roots real what this means

Recurrence

\(P_{n}(x) = \left(1 - 22\,x + 8\,n x\right)\,P_{n-2}(x) + \left(2\,x - 4\,x^{2}\right)\,P_{n-2}'(x) + \left(-6\,x + 8\,n x\right)\,P_{n-1}(x) + \left(2\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{2}(x) = 2 + 18\,x + 12\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	P_n(1)
1	2	2															4
2	2	18	12														32
3	2	82	296	120													500
4	2	290	3156	6072	1680												11200
5	2	930	23952	122288	147744	30240											325156
6	2	2866	153308	1647408	4950720	4160544	665280										11580128
7	2	8690	892728	17802152	106227296	212921952	133223040	17297280									488373140
8	2	26178	4915620	168193640	1762696080	6765250656	9789891264	4783720320	518918400								23793612160
9	2	78658	26148896	1456377440	24816552640	161741180096	436799783424	481870775808	190464606720	17643225600							1314818729284
10	2	236114	136093356	11887228256	312428994560	3205222846656	14359969237632	28990792140288	25365822617088	8330583421440	670442572800						81247285388192
11	2	708498	698411592	93105555928	3633531289920	55692522720576	384161938813184	1264322958287616	1992904925243904	1425052650355200	397144985610240	28158588057600					5551165905054260
12	2	2125666	3551571956	708095845400	39879352784656	879184875657536	8868181490007168	44208069721682688	112039297640678400	142463079005465088	85226384295429120	20499533265254400	1295295050649600				415519616347151680
13	2	6377186	17953421232	5271637897744	419150235505888	12914386499978208	183441639794620928	1316404643189566976	4989346466245320192	10085650313002089984	10611131050861572096	5411447757891440640	1139131903885516800	64764752532480000			33814657353735787876
14	2	19131762	90404824956	38637162485392	4262881497438144	179489491933609440	3488766217199697728	34720300491978567168	187154674011347206656	559958421900454497792	927700903263876867072	824119308510347759616	363824287588989665280	67807936383331737600	3497296636753920000		2972455686105297408608
15	2	57395506	454090355288	279923159552776	42269755531774112	2389085876896275616	62129270312484032384	833393917428318189184	6157387841441152214528	25946859085857882748416	63078919044987210762240	87520400528136013191168	66740830974383370854400	25834114570352595886080	4304995540212171571200	202843204931727360000	280684305614052662162900

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Technical details

Node id: n-576ec26b3dfabfc3 · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-58f7968dbeed8539

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 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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1: 2,      2;
   2: 2,     18,        12;
   3: 2,     82,       296,         120;
   4: 2,    290,      3156,        6072,         1680;
   5: 2,    930,     23952,      122288,       147744,         30240;
   6: 2,   2866,    153308,     1647408,      4950720,       4160544,         665280;
   7: 2,   8690,    892728,    17802152,    106227296,     212921952,      133223040,       17297280;
   8: 2,  26178,   4915620,   168193640,   1762696080,    6765250656,     9789891264,     4783720320,      518918400;
   9: 2,  78658,  26148896,  1456377440,  24816552640,  161741180096,   436799783424,   481870775808,   190464606720,   17643225600;
  10: 2, 236114, 136093356, 11887228256, 312428994560, 3205222846656, 14359969237632, 28990792140288, 25365822617088, 8330583421440, 670442572800;

%F Formula

The row polynomials satisfy P(n,x) = (1+(8*n-22)*x)*P(n-2,x) + (2*x-4*x^2)*d/dx P(n-2,x) + (8*n-6)*x*P(n-1,x) + (2*x-4*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 2 + 2*x, P(2,x) = 2 + 18*x + 12*x^2.

%t Mathematica

P[1, x_] := 2 + 2*x;
P[2, x_] := 2 + 18*x + 12*x^2;
P[n_, x_] := P[n, x] = Expand[(1 + (8*n - 22)*x)*P[n - 2, x] + (2*x - 4*x^2)*D[P[n - 2, x], x] + (8*n - 6)*x*P[n - 1, x] + (2*x - 4*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] = 2 + 2*x; v[2] = 2 + 18*x + 12*x^2; for(n = 3, N, my(i = n); v[i] = (1 + (8*n - 22)*x)*v[i - 2] + (2*x - 4*x^2)*deriv(v[i - 2], x) + (8*n - 6)*x*v[i - 1] + (2*x - 4*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-8644a9f1d4c74332

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-8644a9f1d4c74332

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): Triangle T(n, k) read by rows: row n gives the coefficients of the numerator polynomials of the o.g.f. of the (n+1)-th …
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check liu_wang e-f3090409c727b3ad root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
- premise (sequence): Triangle T(n, k) read by rows: row n gives the coefficients of the numerator polynomials of the o.g.f. of the (n+1)-th …
  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.