A128813 → Pₙ + Pₙ₊₁

nonnegative coefficients only rational roots

Proved for all n: all roots real what this means

Recurrence

\(\left(8 + 2\,n x + 2\,n^{2} x\right)\,P_{n}(x) = \left(8 + 4\,x + 10\,n x + 2\,n x^{2} + 6\,n^{2} x + 5\,n^{2} x^{2} + 4\,n^{3} x^{2} + n^{4} x^{2}\right)\,P_{n-1}(x)\)

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

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	2	5	3														10
2	2	14	30	18													64
3	2	30	154	306	180												672
4	2	55	554	2481	4680	2700											10472
5	2	91	1589	13353	55053	99900	56700										226688
6	2	140	3892	54856	413586	1609884	2835000	1587600									6504960
7	2	204	8484	185672	2299122	16094556	60013224	103194000	57153600								238948864
8	2	285	16908	542882	10261098	116287749	770654088	2778784380	4686595200	2571912000							10935054592
9	2	385	31383	1415282	38699848	661287249	7020661563	44592409980	156503672100	259763112000	141455160000						610036449792
10	2	506	54978	3363866	127698494	3123902118	49546855350	500108462334	3070153773000	10538081613000	17257529520000	9336040560000					40754715803648
11	2	650	91806	7409402	377907530	12719886582	286496244090	4289076270846	41575631051052	248187585141000	836141600934000	1353725881200000	728211163680000				3212430539816960
12	2	819	147238	15312011	1021619742	45844264297	1410669753354	29798119514673	425925059703696	3993632376734232	23287560273396000	77220102599514000	123795897825600000	66267215894880000			295021589700440064
13	2	1015	228137	29973203	2558542129	149186162957	6082139582835	174512850780897	3501095246177193	48174560724502584	439288266104298360	2510947573934058000	8213085213005610000	13054641531291360000	6958057668962400000		31227876656033677312
14	2	1240	343112	55996928	6001545004	445095680432	23449148660760	886858403887872	24053128792772418	462496974642838584	6163899667814514960	54888511485247531200	308388658393709460000	996778851792035760000	1572521033185502400000	834966920275488000000	3774195335661939392512

Computed in your browser ?

Technical details

Node id: n-9f660947dd0578ca · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-f1db139b52e8ecd3

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: 2,   5,     3;
  2: 2,  14,    30,      18;
  3: 2,  30,   154,     306,      180;
  4: 2,  55,   554,    2481,     4680,      2700;
  5: 2,  91,  1589,   13353,    55053,     99900,      56700;
  6: 2, 140,  3892,   54856,   413586,   1609884,    2835000,     1587600;
  7: 2, 204,  8484,  185672,  2299122,  16094556,   60013224,   103194000,     57153600;
  8: 2, 285, 16908,  542882, 10261098, 116287749,  770654088,  2778784380,   4686595200,   2571912000;
  9: 2, 385, 31383, 1415282, 38699848, 661287249, 7020661563, 44592409980, 156503672100, 259763112000, 141455160000;

%F Formula

The row polynomials satisfy (8+(2*n^2+2*n)*x)*P(n,x) = (8+(6*n^2+10*n+4)*x+(n^4+4*n^3+5*n^2+2*n)*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[((8 + (6*n^2 + 10*n + 4)*x + (n^4 + 4*n^3 + 5*n^2 + 2*n)*x^2)*P[n - 1, x])/(8 + (2*n^2 + 2*n)*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] = ((8 + (6*n^2 + 10*n + 4)*x + (n^4 + 4*n^3 + 5*n^2 + 2*n)*x^2)*v[i - 1])/(8 + (2*n^2 + 2*n)*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-4ea38c3ef421f3f7

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-4ea38c3ef421f3f7

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check liu_wang e-fd268ef2c64764c5 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - parent_provers paper_theorem e-a464fe87e26690eb root: theorem check
    
    Parent-row provers of polytool polytool ?
- premise (sequence): Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).
  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.