A144084 → Pₙ + Pₙ₊₁

nonnegative coefficients

Proved for all n: all roots real what this means

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	P_n(1)
0	2	1								3
1	2	5	2							9
2	2	13	20	6						41
3	2	25	90	102	24					243
4	2	41	272	696	624	120				1755
5	2	61	650	3000	6000	4440	720			14873
6	2	85	1332	9750	34800	57240	36000	5040		144249
7	2	113	2450	26166	147000	429240	599760	327600	40320	1572651

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-094e4df3b51dddea · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-164505adeccf5424

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,   5,    2;
  2: 2,  13,   20,      6;
  3: 2,  25,   90,    102,      24;
  4: 2,  41,  272,    696,     624,     120;
  5: 2,  61,  650,   3000,    6000,    4440,      720;
  6: 2,  85, 1332,   9750,   34800,   57240,    36000,     5040;
  7: 2, 113, 2450,  26166,  147000,  429240,   599760,   327600,    40320;
  8: 2, 145, 4160,  61152,  498624, 2281440,  5644800,  6854400,  3306240,   362880;
  9: 2, 181, 6642, 128736, 1439424, 9525600, 36832320, 79107840, 84913920, 36650880, 3628800;

%F Formula

The row polynomials satisfy (2+(2*n-1)*x+n^2*x^2)*P(n,x) = (2+(6*n-1)*x+(5*n^2-1)*x^2+(2*n^3+n^2)*x^3)*P(n-1,x) - (2*x^2+(2*n+1)*x^3+n^2*x^4)*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[Together[((2 + (6*n - 1)*x + (5*n^2 - 1)*x^2 + (2*n^3 + n^2)*x^3)*P[n - 1, x] - (2*x^2 + (2*n + 1)*x^3 + n^2*x^4)*D[P[n - 1, x], x])/(2 + (2*n - 1)*x + n^2*x^2)]];
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] = ((2 + (6*n - 1)*x + (5*n^2 - 1)*x^2 + (2*n^3 + n^2)*x^3)*v[i - 1] - (2*x^2 + (2*n + 1)*x^3 + n^2*x^4)*deriv(v[i - 1], x))/(2 + (2*n - 1)*x + n^2*x^2)); 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-50ecd620d262c41e

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-50ecd620d262c41e

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- 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, +inf)
  - theorem_check liu_wang e-c98efb5092e3dc49 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
- premise (sequence): T(n,k) is the number of partial bijections of height k (height(alpha) = |Im(alpha)|) of an n-element set.
  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.