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	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	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
8	36486450	326351025	1297296000	3008105100	4483779300	4455400950	2951348400	1256755500	312161850	34459425							18162144000
9	689188500	6857425575	30703347675	81462080700	141834993300	169333614450	140387697450	79808028300	29772943200	6581750175	654729075						688085798400
10	14404039650	157789707075	785674890000	2347203733875	4674765595500	6517173212550	6489674591400	4615839978750	2298099053250	762759372375	151897145400	13749310575					28829030630400
11	329983453800	3946052135025	21627665534475	71840147754375	161073173386125	256809622919850	298552529825550	254994713923950	158804537141250	70327723591125	21022695869175	3808559029275	316234143225				1323453638707200
12	8222087723850	106570906266825	637528032741600	2330645635568250	5809221211043250	10425290999698575	13858645092692400	13816902185786700	10331369459160750	5722256821656375	2281945577511600	620451389007450	103092330691350	7905853580625			66060047583129600
13	221363900257500	3091188750024375	20041338826884375	79959803114441250	219324190033698750	437517843005368125	654580958915008125	746154460939387500	651189347728920000	432995694757250625	215932578847610625	78315385569671250	19527458344143750	2996318507056875	213458046676875		3562061389286400000
14	6403741400306250	95842662957916875	669404434378680000	2894277654891748125	8663408282427648750	19016763920396116875	31623382699085677500	40567061396800063125	40475487894775683750	31409711194362125625	18803092415672565000	8527435506694479375	2836003608148961250	652968164784560625	93067708351117500	6190283353629375	206340501568481280000

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.