A142963 → Pₙ + Pₙ₊₁

nonnegative coefficients

Proved for all n: all roots real what this means

Recurrence

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

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

\(P_{1}(x) = 2 + 12\,x + 4\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	P_n(1)
0	2	2															4
1	2	12	4														18
2	2	40	76	8													126
3	2	104	588	472	16												1182
4	2	240	3100	7480	2880	32											13734
5	2	520	13324	71256	87344	17408	64										189918
6	2	1088	50748	514520	1406240	966432	104768	128									3043926
7	2	2232	179116	3131576	16035600	25148032	10328192	629376	256								55454382
8	2	4528	601372	17023288	147271552	434241952	420573568	107898112	3778048	512							1131392934
9	2	9128	1952076	85577880	1168238064	5786570496	10663574208	6710320896	1110466560	22672384	1024						25549382718
10	2	18336	6190972	406833048	8351158752	64563381792	200309398464	243985502592	103509616128	11316278272	136043520	2048					632584423926
11	2	36760	19318636	1856629880	55303173840	633772331904	3059307605760	6317677890432	5296913383680	1557870653440	114568573952	816281600	4096				17038105883982
12	2	73616	59597916	8219097592	345725416000	5653679988384	40191575780352	129323196175872	185711604240384	110513535731200	23025269036032	1154936948736	4897734656	8192			495932699828934
13	2	147336	182376460	35559424600	2067745047984	46867694956544	471267808156992	2228855837091840	5009486356976640	5170012608775168	2236335294369792	335755279763456	11609774919680	29386506240	16384		15512323528529118
14	2	294784	554889148	151176599640	11947666512032	366860510872352	5059530687404608	33704662876676736	111458080722600960	181224702994008064	137904097796384768	44191455309799424	4846884507967488	116489569361920	176319250432	32768	518885040692655126

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Technical details

Node id: n-5254037deeca106e · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-a71ac548f77bbfdd

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): 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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0: 2,    2;
  1: 2,   12,       4;
  2: 2,   40,      76,        8;
  3: 2,  104,     588,      472,         16;
  4: 2,  240,    3100,     7480,       2880,         32;
  5: 2,  520,   13324,    71256,      87344,      17408,          64;
  6: 2, 1088,   50748,   514520,    1406240,     966432,      104768,        128;
  7: 2, 2232,  179116,  3131576,   16035600,   25148032,    10328192,     629376,        256;
  8: 2, 4528,  601372, 17023288,  147271552,  434241952,   420573568,  107898112,    3778048,      512;
  9: 2, 9128, 1952076, 85577880, 1168238064, 5786570496, 10663574208, 6710320896, 1110466560, 22672384, 1024;

%F Formula

The row polynomials satisfy P(n,x) = (1+(4*n-6)*x)*P(n-2,x) + (x-4*x^2)*d/dx P(n-2,x) + (4*n+2)*x*P(n-1,x) + (x-4*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 2 + 2*x, P(1,x) = 2 + 12*x + 4*x^2.

%t Mathematica

P[0, x_] := 2 + 2*x;
P[1, x_] := 2 + 12*x + 4*x^2;
P[n_, x_] := P[n, x] = Expand[(1 + (4*n - 6)*x)*P[n - 2, x] + (x - 4*x^2)*D[P[n - 2, x], x] + (4*n + 2)*x*P[n - 1, x] + (x - 4*x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 2 + 2*x; v[2] = 2 + 12*x + 4*x^2; for(n = 2, N, my(i = n + 1); v[i] = (1 + (4*n - 6)*x)*v[i - 2] + (x - 4*x^2)*deriv(v[i - 2], x) + (4*n + 2)*x*v[i - 1] + (x - 4*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 - 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-08a4b5e6d1540539

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-08a4b5e6d1540539

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): Triangle read by rows, coefficients of the polynomials P(k, x) = (1/2) Sum_{p=0..k-1} Stirling2(k, p+1)*x^p*(1-4*x)^(k-…
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check liu_wang e-a23b1eece2788390 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
- premise (sequence): Triangle read by rows, coefficients of the polynomials P(k, x) = (1/2) Sum_{p=0..k-1} Stirling2(k, p+1)*x^p*(1-4*x)^(k-…
  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.