A038326 → 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

\(P_{n}(x) = \left(23 + 48\,x\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	23	48							71
2	529	2208	2304						5041
3	12167	76176	158976	110592					357911
4	279841	2336064	7312896	10174464	5308416				25411681
5	6436343	67161840	280327680	585031680	610467840	254803968			1804229351
6	148035889	1853666784	9671304960	26911457280	42122280960	35162947584	12230590464		128100283921
7	3404825447	49740058704	311416019712	1083186155520	2260562411520	2830617280512	1969125064704	587068342272	9095120158391

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-3c88a49254d309ee · revision 1 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-d0358a63ea8a15aa

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,             48;
  2:           529,           2208,            2304;
  3:         12167,          76176,          158976,           110592;
  4:        279841,        2336064,         7312896,         10174464,          5308416;
  5:       6436343,       67161840,       280327680,        585031680,        610467840,        254803968;
  6:     148035889,     1853666784,      9671304960,      26911457280,      42122280960,      35162947584,       12230590464;
  7:    3404825447,    49740058704,    311416019712,    1083186155520,    2260562411520,    2830617280512,     1969125064704,      587068342272;
  8:   78310985281,  1307452971648,   9550091271168,   39861250523136,  103985870929920,  173611193204736,   181159505952768,   108020574978048,   28179280429056;
  9: 1801152661463, 33830345641392, 282409841875968, 1375213143048192, 4305015056498688, 8984379248345088, 12500005910740992, 11180129510227968, 5833111048814592, 1352605460594688;

%F Formula

The row polynomials satisfy P(n,x) = (23+48*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 + 48*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 + 48*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-c0f65529a8792e3f
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-c0f65529a8792e3f
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-c0f65529a8792e3f

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-c0f65529a8792e3f

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