A038333 → basis A013611

nonnegative coefficients only rational roots

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence

\(P_{n}(x) = \left(19 + 28\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 361 + 1064\,x + 784\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	361	1064	784								2209
1	6859	30324	44688	21952							103823
2	130321	768208	1698144	1668352	614656						4879681
3	2476099	18244940	53774560	79246720	58392320	17210368					229345007
4	47045881	415984632	1532574960	3011375360	3328362240	1961981952	481890304				10779215329
5	893871739	9220992676	40766493936	100128230720	147557392640	130471799808	64091410432	13492928512			506623120463
6	16983563041	200227269536	1032751179712	3043898213888	5607180920320	6610571190272	4870947192832	2050925133824	377801998336		23811286661761
7	322687697779	4279857886332	25228635961536	86751099095808	191765587474944	282601918384128	277643989991424	175354098941952	64604141715456	10578455953408	1119130473102767

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-3afe7d5f29fd0242 · revision 1 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-304eeb06eba1cf88

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (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): yes

OEIS matches ?

A009991: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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:             361,             1064,               784;
  1:            6859,            30324,             44688,             21952;
  2:          130321,           768208,           1698144,           1668352,             614656;
  3:         2476099,         18244940,          53774560,          79246720,           58392320,           17210368;
  4:        47045881,        415984632,        1532574960,        3011375360,         3328362240,         1961981952,          481890304;
  5:       893871739,       9220992676,       40766493936,      100128230720,       147557392640,       130471799808,        64091410432,        13492928512;
  6:     16983563041,     200227269536,     1032751179712,     3043898213888,      5607180920320,      6610571190272,      4870947192832,      2050925133824,       377801998336;
  7:    322687697779,    4279857886332,    25228635961536,    86751099095808,    191765587474944,    282601918384128,    277643989991424,    175354098941952,     64604141715456,     10578455953408;
  8:   6131066257801,   90352555378120,   599180104086480,  2354672689743360,   6072576936706560,  10738872898596864,  13188089524592640,  11105759599656960,   6137393462968320,   2009906631147520,   296196766695424;
  9: 116490258898219, 1888368407402708, 13914293528230480, 61515824019545280, 181309797110238720, 374070739301124096, 551262142127972352, 580275939082076160, 427571744586792960, 210035242954915840, 61905124239343616, 8293509467471872;

%F Formula

The row polynomials satisfy P(n,x) = (19+28*x)*P(n-1,x) for n >= 1, with P(0,x) = 361 + 1064*x + 784*x^2.

%t Mathematica

P[0, x_] := 361 + 1064*x + 784*x^2;
P[n_, x_] := P[n, x] = Expand[(19 + 28*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 361 + 1064*x + 784*x^2; for(n = 1, N, my(i = n + 1); v[i] = (19 + 28*x)*v[i - 1]); 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, 0]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-0d83dd77e9f6bf78
Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
All coefficients of each polynomial have the same sign (after x -> -x when alternating).

derived from the cited steps by a stated theorem: e-0d83dd77e9f6bf78
Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.

derived from the cited steps by a stated theorem: e-0d83dd77e9f6bf78

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

basis_map A013611 e-0d83dd77e9f6bf78

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*7^j.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-de9af033a7745ebd root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - parent_provers periodic_product e-400776ae4466efdf root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

No candidate equalities or related sequences.

References

[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