A126351 → Pₙ + Pₙ₊₁

nonnegative coefficients

Proved for all n: all roots real what this means

Recurrence

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

\(P_{0}(x) = 2 + 2\,x\)

\(P_{1}(x) = 2 + 7\,x + 4\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	P_n(1)
0	2	2															4
1	2	7	4														13
2	2	14	23	8													47
3	2	23	74	73	16												188
4	2	34	180	350	227	32											825
5	2	47	370	1195	1562	697	64										3937
6	2	62	679	3290	7252	6734	2123	128									20270
7	2	79	1148	7798	26082	41643	28394	6433	256								111835
8	2	98	1824	16548	78288	192234	230540	117950	19427	512							657423
9	2	119	2760	32250	205254	720069	1348370	1244315	484922	58537	1024						4097622
10	2	142	4015	58740	484374	2303994	6232193	9130330	6594852	1979054	176123	2048					26965867
11	2	167	5654	101255	1050654	6516213	24100736	51611516	60244162	34487563	8034314	529393	4096				186685725
12	2	194	7748	166738	2126839	16685526	81006068	239329376	413527114	389823434	178551100	32491550	1590227	8192			1355314108
13	2	223	10374	264173	4064060	39367328	243085700	949371280	2284548266	3230398171	2484925170	917387835	131029082	4774777	16384		10289242825
14	2	254	13615	404950	7395206	86722636	664216410	3318234920	10633174266	21147372246	24740455467	15656246970	4686004052	527304974	14332523	32768	81481911259

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Technical details

Node id: n-b2914cf224361a03 · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-62f4a57d6f45f4d2

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 1)
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 ?

A078468: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0: 2,   2;
  1: 2,   7,    4;
  2: 2,  14,   23,     8;
  3: 2,  23,   74,    73,     16;
  4: 2,  34,  180,   350,    227,     32;
  5: 2,  47,  370,  1195,   1562,    697,      64;
  6: 2,  62,  679,  3290,   7252,   6734,    2123,     128;
  7: 2,  79, 1148,  7798,  26082,  41643,   28394,    6433,    256;
  8: 2,  98, 1824, 16548,  78288, 192234,  230540,  117950,  19427,   512;
  9: 2, 119, 2760, 32250, 205254, 720069, 1348370, 1244315, 484922, 58537, 1024;

%F Formula

The row polynomials satisfy P(n,x) = (1+n*x)*P(n-2,x) - x^2*d/dx P(n-2,x) + (n+2)*x*P(n-1,x) - x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 2 + 2*x, P(1,x) = 2 + 7*x + 4*x^2.

%t Mathematica

P[0, x_] := 2 + 2*x;
P[1, x_] := 2 + 7*x + 4*x^2;
P[n_, x_] := P[n, x] = Expand[(1 + n*x)*P[n - 2, x] - x^2*D[P[n - 2, x], x] + (n + 2)*x*P[n - 1, x] - x^2*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 2 + 2*x; v[2] = 2 + 7*x + 4*x^2; for(n = 2, N, my(i = n + 1); v[i] = (1 + n*x)*v[i - 2] - x^2*deriv(v[i - 2], x) + (n + 2)*x*v[i - 1] - 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 - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 0, 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-68c0b23ee0fd1d24

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-68c0b23ee0fd1d24

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): Triangle read by rows: matrix product of the Stirling numbers of the second kind with the binomial coefficients.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check liu_wang e-0c08584d383e1215 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
- premise (sequence): Triangle read by rows: matrix product of the Stirling numbers of the second kind with the binomial coefficients.
  uses: Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
  
  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.