A089231 → Pₙ + Pₙ₊₁

nonnegative coefficients

Proved for all n: all roots real what this means

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	P_n(1)
0	2	2								4
1	2	8	6							16
2	2	18	42	24						86
3	2	32	156	264	120					574
4	2	50	420	1440	1920	720				4552
5	2	72	930	5400	14400	15840	5040			41684
6	2	98	1806	15960	71400	156240	146160	40320		431986
7	2	128	3192	39984	270480	987840	1834560	1491840	362880	4990906

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-69c5ad82a7bc1667 · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-81b517a7a45b4ff7

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,   2;
  1: 2,   8,    6;
  2: 2,  18,   42,     24;
  3: 2,  32,  156,    264,     120;
  4: 2,  50,  420,   1440,    1920,      720;
  5: 2,  72,  930,   5400,   14400,    15840,     5040;
  6: 2,  98, 1806,  15960,   71400,   156240,   146160,     40320;
  7: 2, 128, 3192,  39984,  270480,   987840,  1834560,   1491840,    362880;
  8: 2, 162, 5256,  88704,  846720,  4656960, 14394240,  23224320,  16692480,   3628800;
  9: 2, 200, 8190, 179280, 2298240, 17781120, 82555200, 221356800, 315705600, 203212800, 39916800;

%F Formula

The row polynomials satisfy (2+2*n*x+(n^2+n)*x^2)*P(n,x) = (2+(6*n+2)*x+(5*n^2+5*n)*x^2+(2*n^3+4*n^2+2*n)*x^3)*P(n-1,x) - (2*x^2+(2*n+2)*x^3+(n^2+n)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 2 + 2*x.

%t Mathematica

P[0, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((2 + (6*n + 2)*x + (5*n^2 + 5*n)*x^2 + (2*n^3 + 4*n^2 + 2*n)*x^3)*P[n - 1, x] - (2*x^2 + (2*n + 2)*x^3 + (n^2 + n)*x^4)*D[P[n - 1, x], x])/(2 + 2*n*x + (n^2 + n)*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 + 2*x; for(n = 1, N, my(i = n + 1); v[i] = ((2 + (6*n + 2)*x + (5*n^2 + 5*n)*x^2 + (2*n^3 + 4*n^2 + 2*n)*x^3)*v[i - 1] - (2*x^2 + (2*n + 2)*x^3 + (n^2 + n)*x^4)*deriv(v[i - 1], x))/(2 + 2*n*x + (n^2 + n)*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-a32213ca02c6584d

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-a32213ca02c6584d

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): Triangular array A066667 or A008297 unsigned and transposed.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check liu_wang e-db00758dc4aca02d root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
- premise (sequence): Triangular array A066667 or A008297 unsigned and transposed.
  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.