A163626 → Pₙ + Pₙ₊₁

Proved for all n: all roots real what this means

Recurrence

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

\(P_{1}(x) = 2 - x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	P_n(1)
1	2	-1								1
2	2	-4	2							0
3	2	-10	14	-6						0
4	2	-22	62	-66	24					0
5	2	-46	230	-450	384	-120				0
6	2	-94	782	-2490	3720	-2640	720			0
7	2	-190	2534	-12306	28560	-34440	20880	-5040		0
8	2	-382	7982	-56826	192024	-349440	352800	-186480	40320	0

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-3cd27d08ccf7c44d · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-f40ca4b4bfa77e79

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 0)
Palindromic rows: no
γ-positive (gamma_positive): no
Nonnegative coefficients (nonn): no
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1: 2,    -1;
   2: 2,    -4,     2;
   3: 2,   -10,    14,       -6;
   4: 2,   -22,    62,      -66,      24;
   5: 2,   -46,   230,     -450,     384,      -120;
   6: 2,   -94,   782,    -2490,    3720,     -2640,      720;
   7: 2,  -190,  2534,   -12306,   28560,    -34440,    20880,     -5040;
   8: 2,  -382,  7982,   -56826,  192024,   -349440,   352800,   -186480,    40320;
   9: 2,  -766, 24710,  -251250, 1187424,  -3056760,  4566240,  -3961440,  1854720,   -362880;
  10: 2, -1534, 75662, -1079130, 6942120, -24277680, 50304240, -63655200, 48384000, -20321280, 3628800;

%F Formula

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

%t Mathematica

P[1, x_] := 2 - x;
P[n_, x_] := P[n, x] = Expand[(1 - x)*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 2 - x; for(n = 2, N, my(i = n); v[i] = (1 - 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, v[i])))};
T(10)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 1, 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-aa85cc5c316d6389

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-aa85cc5c316d6389

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): Triangle read by rows: The n-th derivative of the logistic function written in terms of y, where y = 1/(1 + exp(-x)).
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check ma_wang_interval e-61dbccc566bad14f root: theorem check
    
    Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ?
- premise (sequence): Triangle read by rows: The n-th derivative of the logistic function written in terms of y, where y = 1/(1 + exp(-x)).
  uses: Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
  
  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
[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
[Obr63] Nikola Obreschkoff. Verteilung und berechnung der nullstellen reeller polynome. Hochschulbücher für mathematik, VEB Deutscher Verlag der Wissenschaften, 1963.