A038277 → 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

\(P_{n}(x) = \left(18 + 44\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	18	44							62
2	324	1584	1936						3844
3	5832	42768	104544	85184					238328
4	104976	1026432	3763584	6133248	3748096				14776336
5	1889568	23094720	112907520	275996160	337328640	164916224			916132832
6	34012224	498845952	3048503040	9935861760	18215746560	17810952192	7256313856		56800235584
7	612220032	10475764992	76822276608	312979645440	765061355520	1122089988096	914295545856	319277809664	3521614606208

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-16b272df8b69180c · revision 1 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-eadeb359c78bcf5d

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
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 ?

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:            1;
  1:           18,            44;
  2:          324,          1584,           1936;
  3:         5832,         42768,         104544,           85184;
  4:       104976,       1026432,        3763584,         6133248,         3748096;
  5:      1889568,      23094720,      112907520,       275996160,       337328640,        164916224;
  6:     34012224,     498845952,     3048503040,      9935861760,     18215746560,      17810952192,       7256313856;
  7:    612220032,   10475764992,    76822276608,    312979645440,    765061355520,    1122089988096,     914295545856,     319277809664;
  8:  11019960576,  215501451264,  1843734638592,   9013813788672,  27542208798720,   53860319428608,   65829279301632,   45976004591616,   14048223625216;
  9: 198359290368, 4363904388096, 42669287350272, 243372972294144, 892367565078528, 2181342936858624, 3554781082288128, 3724056371920896, 2275812227284992, 618121839509504;

%F Formula

The row polynomials satisfy P(n,x) = (18+44*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(18 + 44*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (18 + 44*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-f0378e56b9f5156d
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-f0378e56b9f5156d
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-f0378e56b9f5156d

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-f0378e56b9f5156d

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*11^j.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-f9e3d73ad0eb8fa6 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - parent_provers periodic_product e-399c4cc80f25ee4f 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