A112493 → Pₙ + Pₙ₊₁

nonnegative coefficients

Proved for all n: all roots real what this means

Recurrence

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

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

\(P_{2}(x) = 2 + 5\,x + 3\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	P_n(1)
1	2	1								3
2	2	5	3							10
3	2	15	28	15						60
4	2	37	155	225	105					524
5	2	83	676	1960	2310	945				5976
6	2	177	2583	13118	28980	28665	10395			83920
7	2	367	9108	75173	274330	488565	415800	135135		1398480
8	2	749	30507	389735	2187185	6196575	9234225	6891885	2027025	26957888

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-ac6633a57506e279 · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-f4e4377f233f97bf

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): 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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1: 2,    1;
   2: 2,    5,      3;
   3: 2,   15,     28,      15;
   4: 2,   37,    155,     225,       105;
   5: 2,   83,    676,    1960,      2310,       945;
   6: 2,  177,   2583,   13118,     28980,     28665,      10395;
   7: 2,  367,   9108,   75173,    274330,    488565,     415800,     135135;
   8: 2,  749,  30507,  389735,   2187185,   6196575,    9234225,    6891885,    2027025;
   9: 2, 1515,  98764, 1887446,  15548940,  65365300,  150930780,  193243050,  128378250,   34459425;
  10: 2, 3049, 312455, 8711516, 101955568, 607937330, 2031259230, 3952008060, 4433779350, 2653375725, 654729075;

%F Formula

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

%t Mathematica

P[1, x_] := 2 + x;
P[2, x_] := 2 + 5*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[(1 + (n - 2)*x)*P[n - 2, x] + (x + x^2)*D[P[n - 2, x], x] + n*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; v[2] = 2 + 5*x + 3*x^2; for(n = 3, N, my(i = n); v[i] = (1 + (n - 2)*x)*v[i - 2] + (x + x^2)*deriv(v[i - 2], x) + n*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-5648409316d57127

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-5648409316d57127

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): Triangle read by rows, T(n, k) = Sum_{j=0..n} C(n-j, n-k)*E2(n, j), where E2 are the second-order Eulerian numbers A201…
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check ma_wang_interval e-6e3ca58296ce0220 root: theorem check
    
    Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ?
- premise (sequence): Triangle read by rows, T(n, k) = Sum_{j=0..n} C(n-j, n-k)*E2(n, j), where E2 are the second-order Eulerian numbers A201…
  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.