A038334 → basis A013609

nonnegative coefficients only rational roots

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

Recurrence

\(P_{n}(x) = \left(20 + 16\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 400 + 640\,x + 256\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	400	640	256								1296
1	8000	19200	15360	4096							46656
2	160000	512000	614400	327680	65536						1679616
3	3200000	12800000	20480000	16384000	6553600	1048576					60466176
4	64000000	307200000	614400000	655360000	393216000	125829120	16777216				2176782336
5	1280000000	7168000000	17203200000	22937600000	18350080000	8808038400	2348810240	268435456			78364164096
6	25600000000	163840000000	458752000000	734003200000	734003200000	469762048000	187904819200	42949672960	4294967296		2821109907456
7	512000000000	3686400000000	11796480000000	22020096000000	26424115200000	21139292160000	11274289152000	3865470566400	773094113280	68719476736	101559956668416

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-7e007679d5feba96 · revision 1 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-6b726760bd05165a

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 ?

A009980: 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:             400,              640,              256;
  1:            8000,            19200,            15360,              4096;
  2:          160000,           512000,           614400,            327680,             65536;
  3:         3200000,         12800000,         20480000,          16384000,           6553600,           1048576;
  4:        64000000,        307200000,        614400000,         655360000,         393216000,         125829120,          16777216;
  5:      1280000000,       7168000000,      17203200000,       22937600000,       18350080000,        8808038400,        2348810240,         268435456;
  6:     25600000000,     163840000000,     458752000000,      734003200000,      734003200000,      469762048000,      187904819200,       42949672960,       4294967296;
  7:    512000000000,    3686400000000,   11796480000000,    22020096000000,    26424115200000,    21139292160000,    11274289152000,     3865470566400,     773094113280,      68719476736;
  8:  10240000000000,   81920000000000,  294912000000000,   629145600000000,   880803840000000,   845571686400000,   563714457600000,   257698037760000,   77309411328000,   13743895347200,   1099511627776;
  9: 204800000000000, 1802240000000000, 7208960000000000, 17301504000000000, 27682406400000000, 31004295168000000, 24803436134400000, 14173392076800000, 5669356830720000, 1511828488192000, 241892558110720, 17592186044416;

%F Formula

The row polynomials satisfy P(n,x) = (20+16*x)*P(n-1,x) for n >= 1, with P(0,x) = 400 + 640*x + 256*x^2.

%t Mathematica

P[0, x_] := 400 + 640*x + 256*x^2;
P[n_, x_] := P[n, x] = Expand[(20 + 16*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] = 400 + 640*x + 256*x^2; for(n = 1, N, my(i = n + 1); v[i] = (20 + 16*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-4eca5e8e987e069c
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-4eca5e8e987e069c
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-4eca5e8e987e069c

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 A013609 e-4eca5e8e987e069c

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-f4f0736bd14a35db root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - parent_provers periodic_product e-b445a5ff4f115ce0 root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (6)

Equal up to content, sign, reversal and a small index shift.

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