A375853 → Pₙ + Pₙ₊₁

nonnegative coefficients

Proved for all n: all roots real what this means

Recurrence

\(\left(2\,n - 5\,n x + 2\,n x^{2} + 4\,n^{2} + n^{2} x^{2}\right)\,P_{n}(x) = \left(12 - 14\,x + 2\,x^{3} + 14\,n + n x - 6\,n x^{2} + 7\,n x^{3} + 4\,n^{2} + 12\,n^{2} x + n^{2} x^{2} + 3\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(8\,x - 12\,x^{2} + 8\,x^{3} - 4\,x^{4} + 8\,n x - 8\,n x^{2} + 2\,n x^{3} - 2\,n x^{4}\right)\,P_{n-1}'(x)\)

\(P_{1}(x) = 10 + 8\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	P_n(1)
1	10	8															18
2	28	64	20														112
3	60	272	236	40													608
4	110	832	1404	656	70												3072
5	182	2072	5764	5192	1526	112											14848
6	280	4480	18616	27456	15496	3136	168										69632
7	408	8736	50760	111280	102440	39744	5880	240									319488
8	570	15744	121992	372640	508300	320960	90984	10272	330								1441792
9	770	26664	265848	1080112	2047820	1912160	881144	190608	16962	440							6422528
10	1012	42944	535788	2795584	7036232	9116352	6204184	2180288	371844	26752	572						28311552
11	1300	66352	1013012	6604312	21315112	36610112	34529992	17906512	4959812	684112	40612	728					123731968
12	1638	99008	1816100	14471600	58290050	128233280	160169240	115018400	46987850	10524800	1198340	59696	910				536870912
13	2030	143416	3112668	29772600	146435250	401662800	642022920	610086960	344934450	113919000	21063900	2013336	85358	1120			2315255808
14	2480	202496	5133232	58051968	342466800	1146172800	2282580720	2773866240	2074156560	947673600	258320400	40100736	3263312	119168	1360		9932111872
15	2992	279616	8187472	108081376	753390768	3022366080	7339507920	11106030240	10614744720	6411524160	2417159280	553122336	73113872	5126656	162928	1632	42412802048

Computed in your browser ?

Technical details

Node id: n-1fefd5af9c8edffe · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-d4d40bef4b30a829

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:   10,     8;
   2:   28,    64,     20;
   3:   60,   272,    236,      40;
   4:  110,   832,   1404,     656,      70;
   5:  182,  2072,   5764,    5192,    1526,     112;
   6:  280,  4480,  18616,   27456,   15496,    3136,     168;
   7:  408,  8736,  50760,  111280,  102440,   39744,    5880,     240;
   8:  570, 15744, 121992,  372640,  508300,  320960,   90984,   10272,    330;
   9:  770, 26664, 265848, 1080112, 2047820, 1912160,  881144,  190608,  16962,   440;
  10: 1012, 42944, 535788, 2795584, 7036232, 9116352, 6204184, 2180288, 371844, 26752, 572;

%F Formula

The row polynomials satisfy (4*n^2+2*n-5*n*x+(n^2+2*n)*x^2)*P(n,x) = (4*n^2+14*n+12+(12*n^2+n-14)*x+(n^2-6*n)*x^2+(3*n^2+7*n+2)*x^3)*P(n-1,x) + ((8*n+8)*x-(8*n+12)*x^2+(2*n+8)*x^3-(2*n+4)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 10 + 8*x.

%t Mathematica

P[1, x_] := 10 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 + 14*n + 12 + (12*n^2 + n - 14)*x + (n^2 - 6*n)*x^2 + (3*n^2 + 7*n + 2)*x^3)*P[n - 1, x] + ((8*n + 8)*x - (8*n + 12)*x^2 + (2*n + 8)*x^3 - (2*n + 4)*x^4)*D[P[n - 1, x], x])/(4*n^2 + 2*n - 5*n*x + (n^2 + 2*n)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 10 + 8*x; for(n = 2, N, my(i = n); v[i] = ((4*n^2 + 14*n + 12 + (12*n^2 + n - 14)*x + (n^2 - 6*n)*x^2 + (3*n^2 + 7*n + 2)*x^3)*v[i - 1] + ((8*n + 8)*x - (8*n + 12)*x^2 + (2*n + 8)*x^3 - (2*n + 4)*x^4)*deriv(v[i - 1], x))/(4*n^2 + 2*n - 5*n*x + (n^2 + 2*n)*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-01a8bab2955d58ac

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-01a8bab2955d58ac

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): Triangle read by rows: T(n, k) = k*(n - k)*binomial(2*n+2, 2*k+1)/(4*n + 2) for 1 <= k <= n-1.
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check liu_wang e-b3e0ac60e2e16f0a root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
- premise (sequence): Triangle read by rows: T(n, k) = k*(n - k)*binomial(2*n+2, 2*k+1)/(4*n + 2) for 1 <= k <= n-1.
  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.