A322944 → Pₙ + Pₙ₊₁

nonnegative coefficients

Proved for all n: all roots real what this means

Recurrence

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

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

\(P_{2}(x) = 3 + 7\,x + x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	P_n(1)
1	2	1								3
2	3	7	1							11
3	8	44	16	1						69
4	30	310	203	29	1					573
5	144	2472	2528	608	46	1				5799
6	840	22120	32620	11620	1435	67	1			68703
7	5760	219840	443840	217960	38840	2908	92	1		929241
8	45360	2404080	6403320	4138120	993370	105658	5299	121	1	14095329

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-7a687a7741c3472f · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-213b5fcf5e8e536c

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:       3,         7,          1;
   3:       8,        44,         16,          1;
   4:      30,       310,        203,         29,         1;
   5:     144,      2472,       2528,        608,        46,         1;
   6:     840,     22120,      32620,      11620,      1435,        67,        1;
   7:    5760,    219840,     443840,     217960,     38840,      2908,       92,      1;
   8:   45360,   2404080,    6403320,    4138120,    993370,    105658,     5299,    121,     1;
   9:  403200,  28694400,   98040320,   80682560,  24961440,   3526320,   248640,   8928,   154,   1;
  10: 3991680, 371226240, 1591349760, 1627077760, 628878320, 113016624, 10483032, 525288, 14163, 191, 1;

%F Formula

The row polynomials satisfy P(n,x) = (n-2+x)*P(n-2,x) + 3*x*d/dx P(n-2,x) + (n-1+x)*P(n-1,x) + 3*x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 2 + x, P(2,x) = 3 + 7*x + x^2.

%t Mathematica

P[1, x_] := 2 + x;
P[2, x_] := 3 + 7*x + x^2;
P[n_, x_] := P[n, x] = Expand[(n - 2 + x)*P[n - 2, x] + 3*x*D[P[n - 2, x], x] + (n - 1 + x)*P[n - 1, x] + 3*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; v[2] = 3 + 7*x + x^2; for(n = 3, N, my(i = n); v[i] = (n - 2 + x)*v[i - 2] + 3*x*deriv(v[i - 2], x) + (n - 1 + x)*v[i - 1] + 3*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-11c2cb91bf9b2413

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-11c2cb91bf9b2413

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): Coefficients of a family of orthogonal polynomials. Triangle read by rows, T(n, k) for 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-d88ecdfd26f75232 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
- premise (sequence): Coefficients of a family of orthogonal polynomials. Triangle read by rows, T(n, k) for 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 ?

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.