A225466 → Pₙ + Pₙ₊₁

nonnegative coefficients

Proved for all n: all roots real what this means

Recurrence

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

\(P_{1}(x) = 3 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	P_n(1)
1	3	3															6
2	6	24	9														39
3	12	138	144	27													321
4	24	726	1566	729	81												3126
5	48	3702	14706	12717	3321	243											34737
6	96	18654	128754	184005	84645	14094	729										430977
7	192	93558	1085994	2410317	1737045	493533	56862	2187									5879688
8	384	468366	8968626	29771469	31549581	13601196	2617839	220887	6561								87204909
9	768	2342982	73154106	354392037	531008541	325869075	93160368	12933918	833247	19683							1393714725
10	1536	11717214	592261794	4117774725	8497295685	7132799898	2840814585	576961218	60466176	3070548	59049						23833222428
11	3072	58590678	4773245994	47072307357	131315463765	146749485321	78214691394	21792551769	3303004230	270444420	11101212	177147					433561066359
12	6144	292962606	38361739986	532115118909	1979633414781	2888687641752	2004542283843	735872764869	151255765287	17751900870	1166572044	39503781	531441				8349720206313
13	12288	1464831462	307772807706	5968351527957	29311213163661	55046590154127	48756908602116	22938700443516	6140268192069	968572421091	90586008018	4882348467	138706101	1594323			169535450812902
14	24576	7324194174	2466576956034	66575185230645	428262038875125	1023725672111142	1140277942504701	673860836007216	228463074324342	46509404787846	5804469519849	442640220399	19917877239	481485546	4782969		3616415568901803
15	49152	36621044598	19754588230794	739726768405197	6195394099943685	18688122542514789	25876735866427446	18919633055680071	7961622440454540	2034161961820560	325271238998706	32905816273512	2084799996279	79494539103	1654907274	14348907	80795530963634613

Computed in your browser ?

Technical details

Node id: n-c9945ec6eb94c8a9 · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-0cf4273626c53359

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:    3,        3;
   2:    6,       24,         9;
   3:   12,      138,       144,         27;
   4:   24,      726,      1566,        729,         81;
   5:   48,     3702,     14706,      12717,       3321,        243;
   6:   96,    18654,    128754,     184005,      84645,      14094,        729;
   7:  192,    93558,   1085994,    2410317,    1737045,     493533,      56862,      2187;
   8:  384,   468366,   8968626,   29771469,   31549581,   13601196,    2617839,    220887,     6561;
   9:  768,  2342982,  73154106,  354392037,  531008541,  325869075,   93160368,  12933918,   833247,   19683;
  10: 1536, 11717214, 592261794, 4117774725, 8497295685, 7132799898, 2840814585, 576961218, 60466176, 3070548, 59049;

%F Formula

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

%t Mathematica

P[1, x_] := 3 + 3*x;
P[n_, x_] := P[n, x] = Expand[(2 + 3*x)*P[n - 1, x] + 3*x*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] = 3 + 3*x; for(n = 2, N, my(i = n); v[i] = (2 + 3*x)*v[i - 1] + 3*x*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-1a132a11ecb2f34a

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-1a132a11ecb2f34a

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): Triangle read by rows, 3^k*S_3(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >=…
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check liu_wang e-0881060763ba73b2 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
- premise (sequence): Triangle read by rows, 3^k*S_3(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >=…
  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.