A227450 → Pₙ + Pₙ₊₁

nonnegative coefficients

Proved for all n: all roots real what this means

Recurrence

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

\(P_{0}(x) = 4 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	P_n(1)
0	4	4								8
1	4	16	8							28
2	4	40	56	16						116
3	4	88	248	176	32					548
4	4	184	920	1200	512	64				2884
5	4	376	3128	6640	4960	1408	128			16644
6	4	760	10136	32816	38080	18368	3712	256		104132
7	4	1528	31928	151536	256032	186368	62720	9472	512	700100

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-9e0bc9f791c813e2 · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-51d435e983c49172

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: 4,    4;
  1: 4,   16,      8;
  2: 4,   40,     56,      16;
  3: 4,   88,    248,     176,      32;
  4: 4,  184,    920,    1200,     512,       64;
  5: 4,  376,   3128,    6640,    4960,     1408,     128;
  6: 4,  760,  10136,   32816,   38080,    18368,    3712,     256;
  7: 4, 1528,  31928,  151536,  256032,   186368,   62720,    9472,    512;
  8: 4, 3064,  98840,  670000, 1583232,  1630272,  811776,  201216,  23552,  1024;
  9: 4, 6136, 302648, 2877680, 9256160, 12948096, 8942976, 3233280, 614400, 57344, 2048;

%F Formula

The row polynomials satisfy P(n,x) = (1+2*x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 4 + 4*x.

%t Mathematica

P[0, x_] := 4 + 4*x;
P[n_, x_] := P[n, x] = Expand[(1 + 2*x)*P[n - 1, x] + x*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] = 4 + 4*x; for(n = 1, N, my(i = n + 1); v[i] = (1 + 2*x)*v[i - 1] + x*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-9dc64aa496a51831

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-9dc64aa496a51831

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): Triangular array read by rows. T(n,k) = A008277(n,k)*2^k; n >= 1, 1 <= k <= n.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check liu_wang e-74361f30d2bc482d root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
- premise (sequence): Triangular array read by rows. T(n,k) = A008277(n,k)*2^k; n >= 1, 1 <= k <= n.
  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.