A038319 → 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

\(P_{n}(x) = \left(16 + 15\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 256 + 480\,x + 225\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	256	480	225								961
1	4096	11520	10800	3375							29791
2	65536	245760	345600	216000	50625						923521
3	1048576	4915200	9216000	8640000	4050000	759375					28629151
4	16777216	94371840	221184000	276480000	194400000	72900000	11390625				887503681
5	268435456	1761607680	4954521600	7741440000	7257600000	4082400000	1275750000	170859375			27512614111
6	4294967296	32212254720	105696460800	198180864000	232243200000	174182400000	81648000000	21870000000	2562890625		852891037441
7	68719476736	579820584960	2174327193600	4756340736000	6688604160000	6270566400000	3919104000000	1574640000000	369056250000	38443359375	26439622160671

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-a875e5d4789de94f · revision 1 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-d935ec63feee39a0

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 ?

A009975: 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:            256,             480,             225;
  1:           4096,           11520,           10800,             3375;
  2:          65536,          245760,          345600,           216000,            50625;
  3:        1048576,         4915200,         9216000,          8640000,          4050000,           759375;
  4:       16777216,        94371840,       221184000,        276480000,        194400000,         72900000,         11390625;
  5:      268435456,      1761607680,      4954521600,       7741440000,       7257600000,       4082400000,       1275750000,        170859375;
  6:     4294967296,     32212254720,    105696460800,     198180864000,     232243200000,     174182400000,      81648000000,      21870000000,       2562890625;
  7:    68719476736,    579820584960,   2174327193600,    4756340736000,    6688604160000,    6270566400000,    3919104000000,    1574640000000,     369056250000,     38443359375;
  8:  1099511627776,  10307921510400,  43486543872000,  108716359680000,  178362777600000,  200658124800000,  156764160000000,   83980800000000,   29524500000000,   6150937500000,    576650390625;
  9: 17592186044416, 181419418583040, 850403524608000, 2391759912960000, 4484549836800000, 5885971660800000, 5518098432000000, 3695155200000000, 1732104000000000, 541282500000000, 101490468750000, 8649755859375;

%F Formula

The row polynomials satisfy P(n,x) = (16+15*x)*P(n-1,x) for n >= 1, with P(0,x) = 256 + 480*x + 225*x^2.

%t Mathematica

P[0, x_] := 256 + 480*x + 225*x^2;
P[n_, x_] := P[n, x] = Expand[(16 + 15*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] = 256 + 480*x + 225*x^2; for(n = 1, N, my(i = n + 1); v[i] = (16 + 15*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-fd6222a072627022
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-fd6222a072627022
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-fd6222a072627022

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-fd6222a072627022

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*11^(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-f6c78a0c1a54b2b2 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - parent_provers periodic_product e-97cca34dfbf5cc39 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