A257615 → Pₙ + Pₙ₊₁

nonnegative coefficients

Proved for all n: all roots real what this means

Recurrence

\(P_{n}(x) = \left(5 + x + 2\,n x\right)\,P_{n-2}(x) + \left(2\,x - 2\,x^{2}\right)\,P_{n-2}'(x) + \left(4 + 5\,x + 2\,n x\right)\,P_{n-1}(x) + \left(2\,x - 2\,x^{2}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 6 + 5\,x\)

\(P_{1}(x) = 30 + 75\,x + 25\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	P_n(1)
0	6	5								11
1	30	75	25							130
2	150	785	740	125						1800
3	750	7095	13585	6505	625					28560
4	3750	59165	198880	192390	53410	3125				510720
5	18750	469155	2546305	4278230	2413080	419495	15625			10160640
6	93750	3596585	29848500	79790475	78058550	27904575	3195840	78125		222566400
7	468750	26926095	328945705	1320702135	2043852885	1272492085	304637715	23824005	390625	5322240000

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-43121871192f6f93 · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-85cf5e3438332e7d

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:        6,          5;
  1:       30,         75,          25;
  2:      150,        785,         740,          125;
  3:      750,       7095,       13585,         6505,          625;
  4:     3750,      59165,      198880,       192390,        53410,          3125;
  5:    18750,     469155,     2546305,      4278230,      2413080,        419495,        15625;
  6:    93750,    3596585,    29848500,     79790475,     78058550,      27904575,      3195840,        78125;
  7:   468750,   26926095,   328945705,   1320702135,   2043852885,    1272492085,    304637715,     23824005,      390625;
  8:  2343750,  198170165,  3465746720,  20064824060,  46229223020,   45506072270,  19121277680,   3188035100,   174814910,    1953125;
  9: 11718750, 1440316155, 35305802185, 285959337340, 939591900520, 1372093120090, 914150652310, 270303806380, 32315937250, 1267844995, 9765625;

%F Formula

The row polynomials satisfy P(n,x) = (5+(2*n+1)*x)*P(n-2,x) + (2*x-2*x^2)*d/dx P(n-2,x) + (4+(2*n+5)*x)*P(n-1,x) + (2*x-2*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 6 + 5*x, P(1,x) = 30 + 75*x + 25*x^2.

%t Mathematica

P[0, x_] := 6 + 5*x;
P[1, x_] := 30 + 75*x + 25*x^2;
P[n_, x_] := P[n, x] = Expand[(5 + (2*n + 1)*x)*P[n - 2, x] + (2*x - 2*x^2)*D[P[n - 2, x], x] + (4 + (2*n + 5)*x)*P[n - 1, x] + (2*x - 2*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] = 6 + 5*x; v[2] = 30 + 75*x + 25*x^2; for(n = 2, N, my(i = n + 1); v[i] = (5 + (2*n + 1)*x)*v[i - 2] + (2*x - 2*x^2)*deriv(v[i - 2], x) + (4 + (2*n + 5)*x)*v[i - 1] + (2*x - 2*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-682ccea737887217

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-682ccea737887217

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 5.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check liu_wang e-e835bed38a18a979 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
- premise (sequence): Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 5.
  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.