A144084 → x ↦ −x (shift -1) → Pₙ + Pₙ₊₁

Proved for all n: all roots real what this means

Recurrence

\(\left(-3 + x + 2\,n - 2\,n x + n^{2} x\right)\,P_{n}(x) = \left(1 + 2\,x - x^{2} - 2\,n - 6\,n x + 4\,n x^{2} + 3\,n^{2} x - 5\,n^{2} x^{2} + 2\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(x^{2} - x^{3} - 2\,n x^{2} + 2\,n x^{3} - n^{2} x^{3}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	P_n(1)
1	1								1
2	-3	2							-1
3	5	-16	6						-5
4	-7	54	-90	24					-19
5	9	-128	504	-576	120				-71
6	-11	250	-1800	4800	-4200	720			-241
7	13	-432	4950	-24000	48600	-34560	5040		-389
8	-15	686	-11466	88200	-323400	529200	-317520	40320	6005

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-deb7177d421fbe93 · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-328f070cb3b06156

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: 0,   1;
   2: 0,  -3,     2;
   3: 0,   5,   -16,      6;
   4: 0,  -7,    54,    -90,      24;
   5: 0,   9,  -128,    504,    -576,      120;
   6: 0, -11,   250,  -1800,    4800,    -4200,      720;
   7: 0,  13,  -432,   4950,  -24000,    48600,   -34560,      5040;
   8: 0, -15,   686, -11466,   88200,  -323400,   529200,   -317520,    40320;
   9: 0,  17, -1024,  23520, -263424,  1528800, -4515840,   6209280, -3225600,    362880;
  10: 0, -19,  1458, -44064,  677376, -5715360, 26671680, -66044160, 78382080, -35925120, 3628800;

%F Formula

The row polynomials satisfy (2*n-3+(n^2-2*n+1)*x)*P(n,x) = -(2*n-1-(3*n^2-6*n+2)*x-(2*n^3-5*n^2+4*n-1)*x^2)*P(n-1,x) - ((2*n-1)*x^2+(n^2-2*n+1)*x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = x.

%t Mathematica

P[1, x_] := x;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 1 - (3*n^2 - 6*n + 2)*x - (2*n^3 - 5*n^2 + 4*n - 1)*x^2)*P[n - 1, x] - ((2*n - 1)*x^2 + (n^2 - 2*n + 1)*x^3)*D[P[n - 1, x], x])/(2*n - 3 + (n^2 - 2*n + 1)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = x; for(n = 2, N, my(i = n); v[i] = (-(2*n - 1 - (3*n^2 - 6*n + 2)*x - (2*n^3 - 5*n^2 + 4*n - 1)*x^2)*v[i - 1] - ((2*n - 1)*x^2 + (n^2 - 2*n + 1)*x^3)*deriv(v[i - 1], x))/(2*n - 3 + (n^2 - 2*n + 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-1c58af07f6ac5587

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-1c58af07f6ac5587

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): Triangle of coefficients of Laguerre polynomials L_n(x) (powers of x in decreasing order).
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)
  - preserver_image affine, shift -1 e-e31dcd4b09a7ff12
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): T(n,k) is the number of partial bijections of height k (height(alpha) = |Im(alpha)|) of an n-element set.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    theorem_check liu_wang e-c98efb5092e3dc49 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
- premise (sequence): Triangle of coefficients of Laguerre polynomials L_n(x) (powers of x in decreasing order).
  uses: Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
  - ↵ preserver_image affine, shift -1 e-e31dcd4b09a7ff12 (expanded above)

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.