A085881 → Pₙ + Pₙ₊₁

nonnegative coefficients only rational roots

Proved for all n: all roots real what this means

Recurrence

\(\left(-x + 2\,n + 2\,n x\right)\,P_{n}(x) = \left(-2 - 3\,x - x^{2} + 2\,n + 2\,n x + 4\,n^{2} + 8\,n^{2} x + 4\,n^{2} x^{2}\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	P_n(1)
0	2	1								3
1	4	7	3							14
2	18	51	48	15						132
3	120	465	675	435	105					1800
4	1050	5145	10080	9870	4830	945				31920
5	11340	67095	165375	217350	160650	63315	10395			695520
6	145530	1008315	2993760	4937625	4885650	2900205	956340	135135		17962560
7	2162160	17162145	59594535	118243125	146621475	116351235	57702645	16351335	2027025	536215680

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-23845d0ec7b5c72c · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-9dac7933332faf05

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:         4,          7,           3;
  2:        18,         51,          48,          15;
  3:       120,        465,         675,         435,          105;
  4:      1050,       5145,       10080,        9870,         4830,          945;
  5:     11340,      67095,      165375,      217350,       160650,        63315,        10395;
  6:    145530,    1008315,     2993760,     4937625,      4885650,      2900205,       956340,      135135;
  7:   2162160,   17162145,    59594535,   118243125,    146621475,    116351235,     57702645,    16351335,     2027025;
  8:  36486450,  326351025,  1297296000,  3008105100,   4483779300,   4455400950,   2951348400,  1256755500,   312161850,   34459425;
  9: 689188500, 6857425575, 30703347675, 81462080700, 141834993300, 169333614450, 140387697450, 79808028300, 29772943200, 6581750175, 654729075;

%F Formula

The row polynomials satisfy (2*n+(2*n-1)*x)*P(n,x) = (4*n^2+2*n-2+(8*n^2+2*n-3)*x+(4*n^2-1)*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[((4*n^2 + 2*n - 2 + (8*n^2 + 2*n - 3)*x + (4*n^2 - 1)*x^2)*P[n - 1, x])/(2*n + (2*n - 1)*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] = ((4*n^2 + 2*n - 2 + (8*n^2 + 2*n - 3)*x + (4*n^2 - 1)*x^2)*v[i - 1])/(2*n + (2*n - 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-7d570c2107f82786

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-7d570c2107f82786

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check liu_wang e-1d211cea19207c02 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - parent_provers root_domain e-088407700c450d53 root: theorem check
    
    Parent-row provers of polytool polytool ?
- premise (sequence): Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).
  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.