A144699 → Pₙ + Pₙ₊₁

nonnegative coefficients

Proved for all n: all roots real what this means

Recurrence

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

\(P_{0}(x) = 2 + 5\,x\)

\(P_{1}(x) = 2 + 21\,x + 25\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	P_n(1)
0	2	5								7
1	2	21	25							48
2	2	55	196	125						378
3	2	125	957	1651	625					3360
4	2	267	3832	13132	12906	3125				33264
5	2	553	13813	82398	154928	95561	15625			362880
6	2	1127	46788	450863	1422218	1645581	679616	78125		4324320
7	2	2277	152389	2260565	11098065	21108583	16183943	4687071	390625	55883520

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-009ca0264e18fd05 · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-d82797a2fb985b5a

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,    5;
  1: 2,   21,      25;
  2: 2,   55,     196,      125;
  3: 2,  125,     957,     1651,       625;
  4: 2,  267,    3832,    13132,     12906,       3125;
  5: 2,  553,   13813,    82398,    154928,      95561,      15625;
  6: 2, 1127,   46788,   450863,   1422218,    1645581,     679616,      78125;
  7: 2, 2277,  152389,  2260565,  11098065,   21108583,   16183943,    4687071,     390625;
  8: 2, 4579,  483736, 10682518,  77715904,  225254512,  280603120,  150120178,   31559926,   1953125;
  9: 2, 9185, 1509213, 48418536, 504206724, 2118867618, 3971957010, 3430266432, 1330271634, 208500181, 9765625;

%F Formula

The row polynomials satisfy P(n,x) = (1+(n+3)*x)*P(n-2,x) + (x-x^2)*d/dx P(n-2,x) + (n+5)*x*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 2 + 5*x, P(1,x) = 2 + 21*x + 25*x^2.

%t Mathematica

P[0, x_] := 2 + 5*x;
P[1, x_] := 2 + 21*x + 25*x^2;
P[n_, x_] := P[n, x] = Expand[(1 + (n + 3)*x)*P[n - 2, x] + (x - x^2)*D[P[n - 2, x], x] + (n + 5)*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 + 5*x; v[2] = 2 + 21*x + 25*x^2; for(n = 2, N, my(i = n + 1); v[i] = (1 + (n + 3)*x)*v[i - 2] + (x - x^2)*deriv(v[i - 2], x) + (n + 5)*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-ab934b884ec7d157

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-ab934b884ec7d157

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): Triangle of 5-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-70c313e2c70fcf1b root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
- premise (sequence): Triangle of 5-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.