A114655 → Pₙ + Pₙ₊₁

nonnegative coefficients

Proved for all n: all roots real what this means

Recurrence

\(\left(-6 + 4\,x - 2\,x^{2} + 15\,n - 2\,n x + n x^{2} + 9\,n^{2} - 2\,n^{2} x + n^{2} x^{2}\right)\,P_{n}(x) = \left(-12 + 10\,x - 8\,x^{2} + 30\,n - 21\,n x + 12\,n x^{2} - 3\,n x^{3} + 18\,n^{2} + 23\,n^{2} x - 4\,n^{2} x^{2} + 3\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(20\,x^{2} - 14\,x^{3} + 2\,x^{4} + 36\,n x - 26\,n x^{2} + 8\,n x^{3} - 2\,n x^{4}\right)\,P_{n-1}'(x)\)

\(P_{1}(x) = 6 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	P_n(1)
1	6	2															8
2	12	14	2														28
3	24	60	26	2													112
4	48	208	184	42	2												484
5	96	640	960	440	62	2											2200
6	192	1824	4160	3200	900	86	2										10364
7	384	4928	15904	18480	8680	1652	114	2									50144
8	768	12800	55552	90944	64288	20384	2800	146	2								247684
9	1536	32256	181248	397824	395136	188160	43008	4464	182	2							1243816
10	3072	79360	560640	1589760	2112768	1411200	483840	83520	6780	222	2						6331164
11	6144	191488	1661440	5913600	10137600	9047808	4346496	1124640	151800	9900	266	2					32591184
12	12288	454656	4753408	20753920	44605440	51296256	32896512	11918016	2410320	261360	13992	314	2				169376484
13	24576	1064960	13205504	69408768	182820352	263357952	217451520	105101568	29748576	4832256	430144	19240	366	2			887465784
14	49152	2465792	35782656	222896128	706241536	1245916672	1287796224	800946432	302318016	68708640	9161152	681408	25844	422	2		4682990076
15	98304	5652480	94904320	691425280	2594801664	5503353856	6962475520	5418167040	2630559360	797020224	148660512	16562000	1044680	34020	482	2	24864759744

Computed in your browser ?

Technical details

Node id: n-68458fe983bd99b5 · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-8abd41dc566391f2

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:    6,     2;
   2:   12,    14,      2;
   3:   24,    60,     26,       2;
   4:   48,   208,    184,      42,       2;
   5:   96,   640,    960,     440,      62,       2;
   6:  192,  1824,   4160,    3200,     900,      86,      2;
   7:  384,  4928,  15904,   18480,    8680,    1652,    114,     2;
   8:  768, 12800,  55552,   90944,   64288,   20384,   2800,   146,    2;
   9: 1536, 32256, 181248,  397824,  395136,  188160,  43008,  4464,  182,   2;
  10: 3072, 79360, 560640, 1589760, 2112768, 1411200, 483840, 83520, 6780, 222, 2;

%F Formula

The row polynomials satisfy (9*n^2+15*n-6-(2*n^2+2*n-4)*x+(n^2+n-2)*x^2)*P(n,x) = (18*n^2+30*n-12+(23*n^2-21*n+10)*x-(4*n^2-12*n+8)*x^2+(3*n^2-3*n)*x^3)*P(n-1,x) + (36*n*x-(26*n-20)*x^2+(8*n-14)*x^3-(2*n-2)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 6 + 2*x.

%t Mathematica

P[1, x_] := 6 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((18*n^2 + 30*n - 12 + (23*n^2 - 21*n + 10)*x - (4*n^2 - 12*n + 8)*x^2 + (3*n^2 - 3*n)*x^3)*P[n - 1, x] + (36*n*x - (26*n - 20)*x^2 + (8*n - 14)*x^3 - (2*n - 2)*x^4)*D[P[n - 1, x], x])/(9*n^2 + 15*n - 6 - (2*n^2 + 2*n - 4)*x + (n^2 + n - 2)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 6 + 2*x; for(n = 2, N, my(i = n); v[i] = ((18*n^2 + 30*n - 12 + (23*n^2 - 21*n + 10)*x - (4*n^2 - 12*n + 8)*x^2 + (3*n^2 - 3*n)*x^3)*v[i - 1] + (36*n*x - (26*n - 20)*x^2 + (8*n - 14)*x^3 - (2*n - 2)*x^4)*deriv(v[i - 1], x))/(9*n^2 + 15*n - 6 - (2*n^2 + 2*n - 4)*x + (n^2 + n - 2)*x^2)); 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-798bfa3a83d1e4b4

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-798bfa3a83d1e4b4

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k weak ascents (1<=k<=n). A Schr…
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check liu_wang e-784ea0037cdd0731 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
- premise (sequence): Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k weak ascents (1<=k<=n). A Schr…
  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.