A225469 → Pₙ + Pₙ₊₁

nonnegative coefficients

Proved for all n: all roots real what this means

Recurrence

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

\(P_{1}(x) = 4 + x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	P_n(1)
1	4	1															5
2	12	11	1														24
3	36	89	22	1													148
4	108	659	331	37	1												1136
5	324	4721	4300	886	56	1											10288
6	972	33371	52021	17590	1950	79	1										105984
7	2916	234569	605602	315871	54640	3767	106	1									1217472
8	8748	1644899	6896191	5343667	1354031	141281	6629	137	1								15395584
9	26244	11523041	77503000	87051196	31070256	4603494	320264	10876	172	1							212108544
10	78732	80687531	864056041	1383270940	677386060	136950618	13250622	657420	16896	211	1						3156355072
11	236196	564891449	9585303982	21613120141	14253606080	3827250274	494717412	33630642	1248780	25125	254	1					50374030336
12	708588	3954476339	106003235251	333782106097	292431635661	102280362382	17184620398	1537267314	77337942	2228655	36047	301	1				857254014976
13	2125764	27682042961	1169990064100	5112734826706	5889983183656	2644879970447	566265113128	64839907132	4244095284	164255487	3778676	50194	352	1			15480789413888
14	6377292	193776426491	12897572748061	77861012464690	117022415316170	66722222503937	17934038024903	2576302234220	213383242072	10650059277	326738555	6137794	68146	407	1		295431712342016
15	19131876	1356441362729	142067076655162	1180812759718411	2301286903471920	1651633532906721	550941249176318	97799407285723	10044715706740	628735553875	24699817142	615214873	9613240	90531	466	1	5936596165705728

Computed in your browser ?

Technical details

Node id: n-63a72abdf1dbaedc · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-2476533376084450

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:     4,        1;
   2:    12,       11,         1;
   3:    36,       89,        22,          1;
   4:   108,      659,       331,         37,         1;
   5:   324,     4721,      4300,        886,        56,         1;
   6:   972,    33371,     52021,      17590,      1950,        79,        1;
   7:  2916,   234569,    605602,     315871,     54640,      3767,      106,      1;
   8:  8748,  1644899,   6896191,    5343667,   1354031,    141281,     6629,    137,     1;
   9: 26244, 11523041,  77503000,   87051196,  31070256,   4603494,   320264,  10876,   172,   1;
  10: 78732, 80687531, 864056041, 1383270940, 677386060, 136950618, 13250622, 657420, 16896, 211, 1;

%F Formula

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

%t Mathematica

P[1, x_] := 4 + x;
P[n_, x_] := P[n, x] = Expand[(3 + x)*P[n - 1, x] + 4*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] = 4 + x; for(n = 2, N, my(i = n); v[i] = (3 + x)*v[i - 1] + 4*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-1f2ac171cb8f6597

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-1f2ac171cb8f6597

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): Triangle read by rows, S_4(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-8ea25153a52b5bd5 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
- premise (sequence): Triangle read by rows, S_4(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.