A038277 → basis A013609

nonnegative coefficients only rational roots

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

Recurrence

\(P_{n}(x) = \left(18 + 22\,x\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	18	22							40
2	324	792	484						1600
3	5832	21384	26136	10648					64000
4	104976	513216	940896	766656	234256				2560000
5	1889568	11547360	28226880	34499520	21083040	5153632			102400000
6	34012224	249422976	762125760	1241982720	1138484160	556592256	113379904		4096000000
7	612220032	5237882496	19205569152	39122455680	47816334720	35065312128	14285867904	2494357888	163840000000

Show all 15 stored rows

Generated b-files: A009984 (row sums) ?

Computed in your browser ?

Technical details

Node id: n-b2ae2bce0e1523fc · revision 1 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-6bae7ffb6b8d3725

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 ?

A009984: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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,            22;
  2:          324,           792,            484;
  3:         5832,         21384,          26136,          10648;
  4:       104976,        513216,         940896,         766656,         234256;
  5:      1889568,      11547360,       28226880,       34499520,       21083040,        5153632;
  6:     34012224,     249422976,      762125760,     1241982720,     1138484160,      556592256,      113379904;
  7:    612220032,    5237882496,    19205569152,    39122455680,    47816334720,    35065312128,    14285867904,     2494357888;
  8:  11019960576,  107750725632,   460933659648,  1126726723584,  1721388049920,  1683134982144,  1028582489088,   359187535872,   54875873536;
  9: 198359290368, 2181952194048, 10667321837568, 30421621536768, 55772972817408, 68166966776832, 55543454410752, 29094190405632, 8889891512832, 1207269217792;

%F Formula

The row polynomials satisfy P(n,x) = (18+22*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 + 22*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 + 22*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-59f75239fe41171d
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-59f75239fe41171d
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-59f75239fe41171d

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 A013609 e-59f75239fe41171d

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 ?

Same fingerprint (3)

Equal up to content, sign, reversal and a small index shift.

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