A109692 → Pₙ + Pₙ₊₁

nonnegative coefficients only rational roots

Proved for all n: all roots real what this means

Recurrence

\(\left(2 - x + 2\,n x\right)\,P_{n}(x) = \left(2 - x - x^{2} + 6\,n x + 4\,n^{2} x^{2}\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	P_n(1)
0	2	1															3
1	2	5	3														10
2	2	13	26	15													56
3	2	25	109	191	105												432
4	2	41	316	1126	1794	945											4224
5	2	61	735	4430	13828	20469	10395										49920
6	2	85	1478	13525	69518	196855	274602	135135									691200
7	2	113	2681	34685	265433	1215347	3188379	4233375	2027025								10967040
8	2	145	4504	78316	834988	5612950	23494616	57907404	73724130	34459425							196116480
9	2	181	7131	160476	2273712	21061614	128064634	498456764	1165768326	1431163845	654729075						3901685760
10	2	221	10770	304635	5536356	67555698	561208660	3146673190	11525742042	25768904481	30640250970	13749310575					85465497600
11	2	265	15653	543675	12329361	191598330	2082016354	15839483990	83009930212	288672829005	620531645193	717165624735	316234143225				2043740160000
12	2	313	22036	922130	25525566	492053199	6769836568	66982463900	473592978782	2343057742663	7788607970196	16170485710770	18217876140450	7905853580625			52973744947200
13	2	365	30199	1498666	49731396	1164363915	19774214317	246511447468	2255844587282	14984911650995	70522565764209	225281808659466	453414157702920	499156041086325	213458046676875		1479340877414400
14	2	421	40446	2348801	92040130	2572402833	52837720078	809680593293	9296986552854	79602375243391	500927489415626	2256040092127731	6955652470034214	13612363330476555	14673171531019050	6190283353629375	44278202813644800

Computed in your browser ?

Technical details

Node id: n-9f409940da7400a5 · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-a4c3164f27ac43ef

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,   5,    3;
  2: 2,  13,   26,     15;
  3: 2,  25,  109,    191,     105;
  4: 2,  41,  316,   1126,    1794,      945;
  5: 2,  61,  735,   4430,   13828,    20469,     10395;
  6: 2,  85, 1478,  13525,   69518,   196855,    274602,    135135;
  7: 2, 113, 2681,  34685,  265433,  1215347,   3188379,   4233375,    2027025;
  8: 2, 145, 4504,  78316,  834988,  5612950,  23494616,  57907404,   73724130,   34459425;
  9: 2, 181, 7131, 160476, 2273712, 21061614, 128064634, 498456764, 1165768326, 1431163845, 654729075;

%F Formula

The row polynomials satisfy (2+(2*n-1)*x)*P(n,x) = (2+(6*n-1)*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[((2 + (6*n - 1)*x + (4*n^2 - 1)*x^2)*P[n - 1, x])/(2 + (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] = ((2 + (6*n - 1)*x + (4*n^2 - 1)*x^2)*v[i - 1])/(2 + (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-fbfe6dbd1940a725

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-fbfe6dbd1940a725

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): Triangle of coefficients in expansion of (1+x)*(1+3x)*(1+5x)*(1+7x)*...*(1+(2n-1)x).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check liu_wang e-e9da8c20b726baa3 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - parent_provers paper_theorem e-63379a8a04b88f9c root: theorem check
    
    Parent-row provers of polytool polytool ?
- premise (sequence): Triangle of coefficients in expansion of (1+x)*(1+3x)*(1+5x)*(1+7x)*...*(1+(2n-1)x).
  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.