A028246 → Pₙ + Pₙ₊₁

nonnegative coefficients

Proved for all n: all roots real what this means

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	P_n(1)
0	2	1								3
1	2	4	2							8
2	2	10	14	6						32
3	2	22	62	66	24					176
4	2	46	230	450	384	120				1232
5	2	94	782	2490	3720	2640	720			10448
6	2	190	2534	12306	28560	34440	20880	5040		103952
7	2	382	7982	56826	192024	349440	352800	186480	40320	1186256

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-f797be9dcc9711d3 · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-d71eb5b8d3097942

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,    4,     2;
  2: 2,   10,    14,       6;
  3: 2,   22,    62,      66,      24;
  4: 2,   46,   230,     450,     384,      120;
  5: 2,   94,   782,    2490,    3720,     2640,      720;
  6: 2,  190,  2534,   12306,   28560,    34440,    20880,     5040;
  7: 2,  382,  7982,   56826,  192024,   349440,   352800,   186480,    40320;
  8: 2,  766, 24710,  251250, 1187424,  3056760,  4566240,  3961440,  1854720,   362880;
  9: 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 >= 1, with P(0,x) = 2 + x.

%t Mathematica

P[0, 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, 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] = (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 - 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-766d0e8c11bad8cd

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-766d0e8c11bad8cd

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): Triangular array a(n,k) = (1/k)*Sum_{i=0..k} (-1)^(k-i)*binomial(k,i)*i^n; n >= 1, 1 <= k <= n, read by rows.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check ma_wang_interval e-994b5c04cf13d0cc root: theorem check
    
    Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ?
- premise (sequence): Triangular array a(n,k) = (1/k)*Sum_{i=0..k} (-1)^(k-i)*binomial(k,i)*i^n; n >= 1, 1 <= k <= n, read by rows.
  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
[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.