A106800 → Pₙ + Pₙ₊₁

nonnegative coefficients

Proved for all n: all roots real what this means

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	2								2
2	2	1							3
3	2	4	1						7
4	2	9	8	1					20
5	2	16	32	16	1				67
6	2	25	90	105	32	1			255
7	2	36	205	440	332	64	1		1080
8	2	49	406	1400	2002	1029	128	1	5017

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-116d755a7dc21756 · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-59e0e9c9ab6dd13a

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (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 ?

A011968: 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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1: 2;
   2: 2,  1;
   3: 2,  4,    1;
   4: 2,  9,    8,    1;
   5: 2, 16,   32,   16,     1;
   6: 2, 25,   90,  105,    32,     1;
   7: 2, 36,  205,  440,   332,    64,     1;
   8: 2, 49,  406, 1400,  2002,  1029,   128,    1;
   9: 2, 64,  728, 3696,  8652,  8736,  3152,  256,   1;
  10: 2, 81, 1212, 8526, 29778, 50295, 37130, 9585, 512, 1;

%F Formula

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

%t Mathematica

P[1, x_] := 2;
P[2, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[(1 + (n - 3)*x)*P[n - 2, x] - x^2*D[P[n - 2, x], x] + (n - 1)*x*P[n - 1, x] - x^2*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; v[2] = 2 + x; for(n = 3, N, my(i = n); v[i] = (1 + (n - 3)*x)*v[i - 2] - x^2*deriv(v[i - 2], x) + (n - 1)*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, 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-e0e912d56bf25b53

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-e0e912d56bf25b53

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): Triangle of Stirling numbers of 2nd kind, S(n, n-k), n >= 0, 0 <= k <= n.
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check liu_wang e-2ab1dfe057e84c9d root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
- premise (sequence): Triangle of Stirling numbers of 2nd kind, S(n, n-k), n >= 0, 0 <= k <= n.
  uses: Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
  
  no recorded property covers this requirement

Related sequences ?

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

A008278 → Pₙ + Pₙ₊₁

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.