A038331 → 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

\(P_{n}(x) = \left(17 + 20\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 289 + 680\,x + 400\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	289	680	400								1369
1	4913	17340	20400	8000							50653
2	83521	393040	693600	544000	160000						1874161
3	1419857	8352100	19652000	23120000	13600000	3200000					69343957
4	24137569	170382840	501126000	786080000	693600000	326400000	64000000				2565726409
5	410338673	3379259660	11926798800	23385880000	27512800000	19420800000	7616000000	1280000000			94931877133
6	6975757441	65654187680	270340772800	636095936000	935435200000	880409600000	517888000000	174080000000	25600000000		3512479453921
7	118587876497	1255636339380	5908876891200	16220446368000	28624317120000	33675667200000	26412288000000	13317120000000	3916800000000	512000000000	129961739795077

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-21ab4a5c6132df39 · revision 1 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-8c1ae502a0a48e51

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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 ?

A009981: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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:            289,             680,              400;
  1:           4913,           17340,            20400,             8000;
  2:          83521,          393040,           693600,           544000,            160000;
  3:        1419857,         8352100,         19652000,         23120000,          13600000,           3200000;
  4:       24137569,       170382840,        501126000,        786080000,         693600000,         326400000,          64000000;
  5:      410338673,      3379259660,      11926798800,      23385880000,       27512800000,       19420800000,        7616000000,        1280000000;
  6:     6975757441,     65654187680,     270340772800,     636095936000,      935435200000,      880409600000,      517888000000,      174080000000,       25600000000;
  7:   118587876497,   1255636339380,    5908876891200,   16220446368000,    28624317120000,    33675667200000,    26412288000000,    13317120000000,     3916800000000,     512000000000;
  8:  2015993900449,  23717575299400,  125563633938000,  393925126080000,   811022318400000,  1144972684800000,  1122522240000000,   754636800000000,   332928000000000,   87040000000000,   10240000000000;
  9: 34271896307633, 443518658098780, 2608933282934000, 9207999822120000, 21665881934400000, 35684982009600000, 41982331776000000, 35279270400000000, 20752512000000000, 8138240000000000, 1914880000000000, 204800000000000;

%F Formula

The row polynomials satisfy P(n,x) = (17+20*x)*P(n-1,x) for n >= 1, with P(0,x) = 289 + 680*x + 400*x^2.

%t Mathematica

P[0, x_] := 289 + 680*x + 400*x^2;
P[n_, x_] := P[n, x] = Expand[(17 + 20*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] = 289 + 680*x + 400*x^2; for(n = 1, N, my(i = n + 1); v[i] = (17 + 20*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-df900b884626232f
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-df900b884626232f
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-df900b884626232f

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-df900b884626232f

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*12^(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-49423b91d6bf9513 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - parent_provers periodic_product e-b8d4501ecbbc70e3 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