A038326 → basis A013610

nonnegative coefficients only rational roots

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

Recurrence

\(P_{n}(x) = \left(23 + 36\,x\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	23	36							59
2	529	1656	1296						3481
3	12167	57132	89424	46656					205379
4	279841	1752048	4113504	4292352	1679616				12117361
5	6436343	50371380	157684320	246810240	193155840	60466176			714924299
6	148035889	1390250088	5440109040	11353271040	13327752960	8344332288	2176782336		42180533641
7	3404825447	37305044028	175171511088	456969159360	715256075520	671718749184	350461956096	78364164096	2488651484819

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-db0f0bf32d2c2b43 · revision 1 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-a30fbdfa808edb9b

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:            23,             36;
  2:           529,           1656,            1296;
  3:         12167,          57132,           89424,           46656;
  4:        279841,        1752048,         4113504,         4292352,          1679616;
  5:       6436343,       50371380,       157684320,       246810240,        193155840,         60466176;
  6:     148035889,     1390250088,      5440109040,     11353271040,      13327752960,       8344332288,       2176782336;
  7:    3404825447,    37305044028,    175171511088,    456969159360,     715256075520,     671718749184,     350461956096,      78364164096;
  8:   78310985281,   980589728736,   5371926340032,  16816465064448,   32901779473920,   41198749949952,   32242499960832,   14419006193664,   2821109907456;
  9: 1801152661463, 25372759231044, 158855536055232, 580168044723456, 1362133670220288, 2132035309910016, 2224732497297408, 1492367141044224, 583969750843392, 101559956668416;

%F Formula

The row polynomials satisfy P(n,x) = (23+36*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[(23 + 36*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] = (23 + 36*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-ffff6b452715bdc2
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-ffff6b452715bdc2
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-ffff6b452715bdc2

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 A013610 e-ffff6b452715bdc2

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*12^j.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-da30941d57c5aa28 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - parent_provers periodic_product e-f0c12876ab8620d1 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