A395972 → Pₙ + Pₙ₊₁

nonnegative coefficients

Proved for all n: all roots real what this means

Recurrence

\(\left(3 + 2\,x + 4\,x^{2} + 4\,n x + 8\,n x^{2} + 4\,n^{2} x^{2}\right)\,P_{n}(x) = \left(6 + 18\,x + 20\,x^{2} + 24\,x^{3} + 20\,n x + 48\,n x^{2} + 64\,n x^{3} + 24\,n^{2} x^{2} + 56\,n^{2} x^{3} + 16\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-6\,x^{2} - 12\,x^{3} - 8\,x^{4} - 8\,n x^{3} - 16\,n x^{4} - 8\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)

\(P_{1}(x) = 12 + 88\,x + 152\,x^{2} + 48\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	P_n(1)
1	12	88	152	48								300
2	24	328	1296	1584	384							3616
3	48	1056	7552	20736	19584	3840						52816
4	96	3104	35200	172800	364800	280320	46080					902400
5	192	8576	141696	1094400	4108800	7050240	4561920	645120				17610944
6	384	22656	514304	5757696	33868800	103111680	149022720	83220480	10321920			385840640
7	768	57856	1728512	26492928	225189888	1071759360	2745630720	3426877440	1682472960	185794560		9366004992
8	1536	143872	5474304	110149632	1278885888	8778792960	35115171840	77662126080	85279703040	37344706560	3715891200	249291046912

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-5482df9a9e2607d0 · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-eddfbac374c8b398

Observed on the stored rows ?

Degrees: 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 (affine: slope 1, intercept 2)
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:   12,     88,      152,         48;
   2:   24,    328,     1296,       1584,         384;
   3:   48,   1056,     7552,      20736,       19584,         3840;
   4:   96,   3104,    35200,     172800,      364800,       280320,         46080;
   5:  192,   8576,   141696,    1094400,     4108800,      7050240,       4561920,         645120;
   6:  384,  22656,   514304,    5757696,    33868800,    103111680,     149022720,       83220480,       10321920;
   7:  768,  57856,  1728512,   26492928,   225189888,   1071759360,    2745630720,     3426877440,     1682472960,      185794560;
   8: 1536, 143872,  5474304,  110149632,  1278885888,   8778792960,   35115171840,    77662126080,    85279703040,    37344706560,     3715891200;
   9: 3072, 350208, 16537600,  423014400,  6436454400,  60259368960,  347250032640,  1198374912000,  2331721728000,  2285273088000,   902961561600,    81749606400;
  10: 6144, 837632, 48074752, 1524019200, 29439590400, 360768798720, 2832623861760, 14073194741760, 42714169344000, 74187767808000, 65644933939200, 23625636249600, 1961990553600;

%F Formula

The row polynomials satisfy (3+(4*n+2)*x+(4*n^2+8*n+4)*x^2)*P(n,x) = (6+(20*n+18)*x+(24*n^2+48*n+20)*x^2+(16*n^3+56*n^2+64*n+24)*x^3)*P(n-1,x) - (6*x^2+(8*n+12)*x^3+(8*n^2+16*n+8)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 12 + 88*x + 152*x^2 + 48*x^3.

%t Mathematica

P[1, x_] := 12 + 88*x + 152*x^2 + 48*x^3;
P[n_, x_] := P[n, x] = Expand[Together[((6 + (20*n + 18)*x + (24*n^2 + 48*n + 20)*x^2 + (16*n^3 + 56*n^2 + 64*n + 24)*x^3)*P[n - 1, x] - (6*x^2 + (8*n + 12)*x^3 + (8*n^2 + 16*n + 8)*x^4)*D[P[n - 1, x], x])/(3 + (4*n + 2)*x + (4*n^2 + 8*n + 4)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 12 + 88*x + 152*x^2 + 48*x^3; for(n = 2, N, my(i = n); v[i] = ((6 + (20*n + 18)*x + (24*n^2 + 48*n + 20)*x^2 + (16*n^3 + 56*n^2 + 64*n + 24)*x^3)*v[i - 1] - (6*x^2 + (8*n + 12)*x^3 + (8*n^2 + 16*n + 8)*x^4)*deriv(v[i - 1], x))/(3 + (4*n + 2)*x + (4*n^2 + 8*n + 4)*x^2)); 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-f9379b86f8cefc86

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-f9379b86f8cefc86

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): Triangle read by rows: T(n, k) = 2^n * [x^k] hypergeom([-n, -n, 1], [1], x).
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check liu_wang e-b9d2d25190d8d499 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
- premise (sequence): Triangle read by rows: T(n, k) = 2^n * [x^k] hypergeom([-n, -n, 1], [1], x).
  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
[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.