A225468 → Pₙ + Pₙ₊₁

nonnegative coefficients

Proved for all n: all roots real what this means

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	P_n(1)
1	3	1															4
2	6	8	1														15
3	12	46	16	1													75
4	24	242	174	27	1												468
5	48	1234	1634	471	41	1											3429
6	96	6218	14306	6815	1045	58	1										28539
7	192	31186	120666	89271	21445	2031	78	1									264870
8	384	156122	996514	1102647	389501	55972	3591	101	1								2704833
9	768	780994	8128234	13125631	6555661	1341025	127792	5914	127	1							30066147
10	1536	3905738	65806866	152510175	104904885	29353086	3896865	263814	9216	156	1						360652338
11	3072	19530226	530360666	1743418791	1621178565	603907347	107290386	9964587	503430	13740	188	1					4636170999
12	6144	97654202	4262415554	19707967367	24439918701	11887603464	2749715067	336475887	23053767	901890	19756	223	1				63505732023
13	12288	488277154	34196978634	221050056591	361866829181	226529177589	66881904804	10488660468	935873829	49208577	1534082	27561	261	1			922488541020
14	24576	2441398058	274064106226	2465747601135	5287185665125	4212862848194	1564167273669	308121095568	34821380022	2362922562	98299201	2498717	37479	302	1		14151875150835
15	49152	12207014866	2194954247866	27397287718711	76486346912885	76905854084423	35496208321574	8650952471733	1213476976140	103346134320	5508496994	185754296	3922919	49861	346	1	228466332156087

Computed in your browser ?

Technical details

Node id: n-5b541b2cce1cbaf5 · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-71b71733ad60bfd3

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,       1;
   2:    6,       8,        1;
   3:   12,      46,       16,         1;
   4:   24,     242,      174,        27,         1;
   5:   48,    1234,     1634,       471,        41,        1;
   6:   96,    6218,    14306,      6815,      1045,       58,       1;
   7:  192,   31186,   120666,     89271,     21445,     2031,      78,      1;
   8:  384,  156122,   996514,   1102647,    389501,    55972,    3591,    101,    1;
   9:  768,  780994,  8128234,  13125631,   6555661,  1341025,  127792,   5914,  127,   1;
  10: 1536, 3905738, 65806866, 152510175, 104904885, 29353086, 3896865, 263814, 9216, 156, 1;

%F Formula

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

%t Mathematica

P[1, x_] := 3 + x;
P[n_, x_] := P[n, x] = Expand[(2 + 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 + x; for(n = 2, N, my(i = n); v[i] = (2 + 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-dfae76bc29223b34

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-dfae76bc29223b34

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): Triangle read by rows, S_3(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check liu_wang e-9f45a63628f73a74 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
- premise (sequence): Triangle read by rows, S_3(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.
  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.