A038324 → 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

\(P_{n}(x) = \left(21 + 40\,x\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	21	40							61
2	441	1680	1600						3721
3	9261	52920	100800	64000					226981
4	194481	1481760	4233600	5376000	2560000				13845841
5	4084101	38896200	148176000	282240000	268800000	102400000			844596301
6	85766121	980184240	4667544000	11854080000	16934400000	12902400000	4096000000		51520374361
7	1801088541	24014513880	137225793600	435637440000	829785600000	948326400000	602112000000	163840000000	3142742836021

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-e9f99a7290620624 · revision 1 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-ae9bc19c6d2e3d8d

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:           21,             40;
  2:          441,           1680,            1600;
  3:         9261,          52920,          100800,           64000;
  4:       194481,        1481760,         4233600,         5376000,          2560000;
  5:      4084101,       38896200,       148176000,       282240000,        268800000,        102400000;
  6:     85766121,      980184240,      4667544000,     11854080000,      16934400000,      12902400000,       4096000000;
  7:   1801088541,    24014513880,    137225793600,    435637440000,     829785600000,     948326400000,     602112000000,     163840000000;
  8:  37822859361,   576348333120,   3842322220800,  14637417984000,   34850995200000,   53106278400000,   50577408000000,   27525120000000,    6553600000000;
  9: 794280046581, 13616229369960, 103742699961600, 461078666496000, 1317367618560000, 2509271654400000, 3186376704000000, 2601123840000000, 1238630400000000, 262144000000000;

%F Formula

The row polynomials satisfy P(n,x) = (21+40*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[(21 + 40*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] = (21 + 40*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-bf7e65720bb019eb
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-bf7e65720bb019eb
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-bf7e65720bb019eb

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-bf7e65720bb019eb

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*10^j.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-0ddfaa48a1d96c8a root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - basis_map A013609 e-0f31cc897fc45a50
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j)*5^j.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    theorem_check liu_wang e-0a7984b7b9125974 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
    
    parent_provers periodic_product e-7949976749bfedcf root: theorem check
    
    Parent-row provers of polytool polytool ?
  - parent_provers periodic_product e-96e2fb6c9c1a89f9 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