A039755 → Pₙ + Pₙ₊₁

nonnegative coefficients

Proved for all n: all roots real what this means

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	P_n(1)
1	2	1															3
2	2	5	1														8
3	2	17	10	1													30
4	2	53	67	17	1												140
5	2	161	388	186	26	1											764
6	2	485	2101	1690	420	37	1										4736
7	2	1457	10990	13931	5470	827	50	1									32728
8	2	4373	56407	108507	63161	14567	1477	65	1								248560
9	2	13121	286408	815956	676956	223398	33768	2452	82	1							2052144
10	2	39365	1445161	5998100	6908560	3134334	662382	70548	3846	101	1						18262400
11	2	118097	7265170	43431861	68175140	41386234	11745300	1720602	135930	5765	122	1					173984224
12	2	354293	36443947	311288197	657008121	523423714	194075134	37554330	4031412	245465	8327	145	1				1764433088
13	2	1062881	182574028	2215461326	6224361286	6414668975	3046400456	757390084	106088334	8695247	420332	11662	170	1			18957134784
14	2	3188645	913933021	15690803310	58234712900	76785720011	46017874903	14407251716	2560891762	271298027	17522219	688558	15912	197	1		214903901184
15	2	9565937	4572853750	110749556191	539803219410	902877633021	675018093750	262126650643	57942411670	7715554275	639264626	33359053	1086358	21231	226	1	2561489270144

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Technical details

Node id: n-a65976744dde69f6 · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-8da1f210cd8a462b

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,     1;
   2: 2,     5,       1;
   3: 2,    17,      10,       1;
   4: 2,    53,      67,      17,       1;
   5: 2,   161,     388,     186,      26,       1;
   6: 2,   485,    2101,    1690,     420,      37,      1;
   7: 2,  1457,   10990,   13931,    5470,     827,     50,     1;
   8: 2,  4373,   56407,  108507,   63161,   14567,   1477,    65,    1;
   9: 2, 13121,  286408,  815956,  676956,  223398,  33768,  2452,   82,   1;
  10: 2, 39365, 1445161, 5998100, 6908560, 3134334, 662382, 70548, 3846, 101, 1;

%F Formula

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

%t Mathematica

P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 1, x] + 2*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] = 2 + x; for(n = 2, N, my(i = n); v[i] = (1 + x)*v[i - 1] + 2*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-a691d30fc1b4b246

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-a691d30fc1b4b246

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): Triangle of B-analogs of Stirling numbers of the second kind.
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check liu_wang e-c873e8cf4f19d199 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
- premise (sequence): Triangle of B-analogs of Stirling numbers of the second kind.
  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.