A144697 → Pₙ + Pₙ₊₁

nonnegative coefficients

Proved for all n: all roots real what this means

Recurrence

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

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

\(P_{1}(x) = 2 + 13\,x + 9\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	P_n(1)
0	2	3															5
1	2	13	9														24
2	2	35	76	27													140
3	2	81	393	403	81												960
4	2	175	1640	3510	1990	243											7560
5	2	365	6089	23554	27124	9337	729										67200
6	2	747	21068	135525	273810	189733	42208	2187									665280
7	2	1513	69681	705869	2297305	2756043	1235027	185599	6561								7257600
8	2	3047	223672	3434312	17018372	32368798	25000808	7618448	799258	19683							86486400
9	2	6117	703521	15920580	115409904	328311054	399011838	209758884	45102150	3387301	59049						1117670400
10	2	12259	2181932	71250767	733411772	2993585798	5389777904	4447346678	1657037618	258591063	14179060	177147					15567552000
11	2	24545	6701081	310653015	4437740265	25195192266	64372199850	78321792078	45842441460	12483826885	1445557357	58782955	531441				232475443200
12	2	49119	20438696	1328077418	25858363770	199353324981	699862253472	1200909561948	1034911286934	444122630905	90548868792	7919062234	241863406	1594323			3705077376000
13	2	98269	62036505	5593399030	146304346004	1502615546391	7069422224967	16546541403780	20048310870690	12676286891227	4092667369607	636890522214	42680929640	989242705	4782969		62768369664000
14	2	196571	187649068	23288757385	808753718302	10895596181363	67340488930608	209495471077929	344773379081310	306206985174841	145997742168028	36202240422371	4368334762178	227012714105	4027121032	14348907	1126343522304000

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Technical details

Node id: n-936d6bd12d8bd41c · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-f1777e18cc7873d6

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): 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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0: 2,    3;
  1: 2,   13,      9;
  2: 2,   35,     76,       27;
  3: 2,   81,    393,      403,        81;
  4: 2,  175,   1640,     3510,      1990,       243;
  5: 2,  365,   6089,    23554,     27124,      9337,       729;
  6: 2,  747,  21068,   135525,    273810,    189733,     42208,      2187;
  7: 2, 1513,  69681,   705869,   2297305,   2756043,   1235027,    185599,     6561;
  8: 2, 3047, 223672,  3434312,  17018372,  32368798,  25000808,   7618448,   799258,   19683;
  9: 2, 6117, 703521, 15920580, 115409904, 328311054, 399011838, 209758884, 45102150, 3387301, 59049;

%F Formula

The row polynomials satisfy P(n,x) = (1+(n+1)*x)*P(n-2,x) + (x-x^2)*d/dx P(n-2,x) + (n+3)*x*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 2 + 3*x, P(1,x) = 2 + 13*x + 9*x^2.

%t Mathematica

P[0, x_] := 2 + 3*x;
P[1, x_] := 2 + 13*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[(1 + (n + 1)*x)*P[n - 2, x] + (x - x^2)*D[P[n - 2, x], x] + (n + 3)*x*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 2 + 3*x; v[2] = 2 + 13*x + 9*x^2; for(n = 2, N, my(i = n + 1); v[i] = (1 + (n + 1)*x)*v[i - 2] + (x - x^2)*deriv(v[i - 2], x) + (n + 3)*x*v[i - 1] + (x - x^2)*deriv(v[i - 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-b4867a0cc7e138f4

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-b4867a0cc7e138f4

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): Triangle of 3-Eulerian numbers.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check liu_wang e-0b8ec5e4abd25b24 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
- premise (sequence): Triangle of 3-Eulerian numbers.
  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.