A156992 → Pₙ + Pₙ₊₁

nonnegative coefficients only rational roots

Proved for all n: all roots real what this means

Recurrence

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

\(P_{0}(x) = 3 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	P_n(1)
0	3	2															5
1	8	14	6														28
2	30	84	78	24													216
3	144	552	792	504	120												2112
4	840	4080	7920	7680	3720	720											24960
5	5760	33840	82800	108000	79200	30960	5040										345600
6	45360	312480	922320	1512000	1486800	876960	287280	40320									5483520
7	403200	3185280	11007360	21732480	26812800	21168000	10442880	2943360	362880								98058240
8	3991680	35562240	140797440	325140480	482630400	477550080	314979840	133539840	33022080	3628800							1950842880
9	43545600	431827200	1926892800	5094835200	8839756800	10516262400	8687347200	4920652800	1828915200	402796800	39916800						42732748800
10	518918400	5668185600	28141344000	83825280000	166453056000	231357772800	229681267200	162860544000	80831520000	26744256000	5308934400	479001600					1021870080000
11	6706022400	79993267200	437328460800	1448979840000	3240445824000	5153099212800	5975065958400	5089871001600	3161410560000	1396289664000	416252390400	75203251200	6227020800				26486872473600
12	93405312000	1208042035200	7210890086400	26302935859200	65414853504000	117130261248000	155351714918400	154529748172800	115280836070400	63702422784000	25343974656000	6874630963200	1139544806400	87178291200			739670438707200
13	1394852659200	19440758937600	125798274201600	500926461235200	1371314520576000	2730162545510400	4076544074803200	4637536378675200	4039144587878400	2680296562944000	1333915033651200	482793376665600	120131685273600	18394619443200	1307674368000		22139101406822400
14	22230464256000	332149289472000	2315891305728000	9995862868992000	29868590239488000	65449102118400000	108645509516544000	139126091360256000	138565099056384000	107336527474176000	64140120076032000	29035601667072000	9638867766528000	2215200379392000	315149522688000	20922789888000	707022915895296000

Computed in your browser ?

Technical details

Node id: n-f455ca73098afb3a · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-38f9189b2b86204b

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:        3,         2;
  1:        8,        14,          6;
  2:       30,        84,         78,         24;
  3:      144,       552,        792,        504,        120;
  4:      840,      4080,       7920,       7680,       3720,         720;
  5:     5760,     33840,      82800,     108000,      79200,       30960,       5040;
  6:    45360,    312480,     922320,    1512000,    1486800,      876960,     287280,      40320;
  7:   403200,   3185280,   11007360,   21732480,   26812800,    21168000,   10442880,    2943360,     362880;
  8:  3991680,  35562240,  140797440,  325140480,  482630400,   477550080,  314979840,  133539840,   33022080,   3628800;
  9: 43545600, 431827200, 1926892800, 5094835200, 8839756800, 10516262400, 8687347200, 4920652800, 1828915200, 402796800, 39916800;

%F Formula

The row polynomials satisfy (n+2+(n+1)*x)*P(n,x) = (n^2+4*n+3+(2*n^2+7*n+5)*x+(n^2+3*n+2)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 3 + 2*x.

%t Mathematica

P[0, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + 4*n + 3 + (2*n^2 + 7*n + 5)*x + (n^2 + 3*n + 2)*x^2)*P[n - 1, x])/(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] = 3 + 2*x; for(n = 1, N, my(i = n + 1); v[i] = ((n^2 + 4*n + 3 + (2*n^2 + 7*n + 5)*x + (n^2 + 3*n + 2)*x^2)*v[i - 1])/(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-107bbdcb60a21a66

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-107bbdcb60a21a66

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check liu_wang e-d07da0cb5a6ac36b root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - parent_provers root_domain e-e811f0fabc9e8e88 root: theorem check
    
    Parent-row provers of polytool polytool ?
- premise (sequence): Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.
  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.