A008955 → Pₙ + Pₙ₊₁

nonnegative coefficients only rational roots

Proved for all n: all roots real what this means

Recurrence

\(\left(2 + n^{2} x\right)\,P_{n}(x) = \left(2 + x + 2\,n x + 3\,n^{2} x + n^{2} x^{2} + 2\,n^{3} x^{2} + n^{4} x^{2}\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	P_n(1)
0	2	1								3
1	2	6	4							12
2	2	19	53	36						110
3	2	44	322	856	576					1800
4	2	85	1296	8465	21652	14400				45900
5	2	146	4026	52118	317372	787536	518400			1679600
6	2	231	10465	236093	2771769	16064776	38920464	25401600		83405400
7	2	344	23884	860808	17214626	189013552	1055469088	2508535296	1625702400	5396820000

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-14e741ac555fb2b8 · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-377ec08a6328063f

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): yes

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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0: 2,   1;
  1: 2,   6,     4;
  2: 2,  19,    53,      36;
  3: 2,  44,   322,     856,       576;
  4: 2,  85,  1296,    8465,     21652,      14400;
  5: 2, 146,  4026,   52118,    317372,     787536,       518400;
  6: 2, 231, 10465,  236093,   2771769,   16064776,     38920464,      25401600;
  7: 2, 344, 23884,  860808,  17214626,  189013552,   1055469088,    2508535296,    1625702400;
  8: 2, 489, 49368, 2668558,  83682534, 1541315217,  16105608920,   87348696336,  204385234176,   131681894400;
  9: 2, 670, 94392, 7293340, 337823762, 9629524710, 166936414292, 1678585314880, 8892075327552, 20539316966400, 13168189440000;

%F Formula

The row polynomials satisfy (2+n^2*x)*P(n,x) = (2+(3*n^2+2*n+1)*x+(n^4+2*n^3+n^2)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 2 + x.

%t Mathematica

P[0, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[Together[((2 + (3*n^2 + 2*n + 1)*x + (n^4 + 2*n^3 + n^2)*x^2)*P[n - 1, x])/(2 + n^2*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 2 + x; for(n = 1, N, my(i = n + 1); v[i] = ((2 + (3*n^2 + 2*n + 1)*x + (n^4 + 2*n^3 + n^2)*x^2)*v[i - 1])/(2 + n^2*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-a0abdef027fe8e86

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-a0abdef027fe8e86

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check liu_wang e-c0780e435a2392b0 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - parent_provers paper_theorem e-b6645f2226a58079 root: theorem check
    
    Parent-row provers of polytool polytool ?
- premise (sequence): Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.
  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.