type_b_eulerian → Pₙ + Pₙ₊₁

nonnegative coefficients

Proved for all n: all roots real what this means

Recurrence

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

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

\(P_{1}(x) = 2 + 7\,x + x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	P_n(1)
0	2	1															3
1	2	7	1														10
2	2	29	24	1													56
3	2	99	253	77	1												432
4	2	313	1912	1758	238	1											4224
5	2	959	12225	25230	10780	723	1										49920
6	2	2901	71200	283271	270266	61379	2180	1									691200
7	2	8731	392269	2745011	4934737	2545945	333791	6553	1								10967040
8	2	26225	2087952	24193260	74088308	71898582	22039584	1762892	19674	1							196116480
9	2	78711	10872481	200008876	976336576	1596505762	928631254	180057244	9135814	59039	1						3901685760
10	2	236173	55818568	1581381629	11743746380	30115112282	29494943096	11015124002	1412196866	46761465	177136	1					85465497600
11	2	708563	283934397	12111991913	132220603305	505608584238	771876534978	487857387522	122786860932	10755613615	237409105	531429	1				2043740160000
12	2	2125737	1435614664	90653160874	1417307114642	7787771036607	17561347678656	17295079727916	7432134252390	1306632245623	80181481656	1198914122	1594310	1			52973744947200
13	2	6377263	7230153889	667116799450	14638062587756	112351549020461	359735586315999	521359834068252	350228277624690	106336721882713	13422083503007	588370932458	6033365504	4782955	1		1479340877414400
14	2	19131845	36319766928	4848174677987	146926361177782	1540674506899077	6796620552552380	13902266360068779	13740642112316526	6558743823948151	1448975847604120	134171394474225	4267033899418	30292778687	14348892	1	44278202813644800

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Technical details

Node id: n-198be0e7a0f207a6 · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-c7ee4b11b3b80bba

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,     1;
  1: 2,     7,        1;
  2: 2,    29,       24,         1;
  3: 2,    99,      253,        77,         1;
  4: 2,   313,     1912,      1758,       238,          1;
  5: 2,   959,    12225,     25230,     10780,        723,         1;
  6: 2,  2901,    71200,    283271,    270266,      61379,      2180,         1;
  7: 2,  8731,   392269,   2745011,   4934737,    2545945,    333791,      6553,       1;
  8: 2, 26225,  2087952,  24193260,  74088308,   71898582,  22039584,   1762892,   19674,     1;
  9: 2, 78711, 10872481, 200008876, 976336576, 1596505762, 928631254, 180057244, 9135814, 59039, 1;

%F Formula

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

%t Mathematica

P[0, x_] := 2 + x;
P[1, x_] := 2 + 7*x + x^2;
P[n_, x_] := P[n, x] = Expand[(1 + (2*n - 3)*x)*P[n - 2, x] + (2*x - 2*x^2)*D[P[n - 2, x], x] + (2*n + 1)*x*P[n - 1, x] + (2*x - 2*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 + x; v[2] = 2 + 7*x + x^2; for(n = 2, N, my(i = n + 1); v[i] = (1 + (2*n - 3)*x)*v[i - 2] + (2*x - 2*x^2)*deriv(v[i - 2], x) + (2*n + 1)*x*v[i - 1] + (2*x - 2*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-6138e831ad364b95

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-6138e831ad364b95

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): type_b_eulerian
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check liu_wang e-4675d8b985e3f46a root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
- premise (sequence): type_b_eulerian
  uses: Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
  
  no recorded property covers this requirement

Related sequences ?

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

A060187 → Pₙ + Pₙ₊₁

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.