A133080 → 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

\(P_{n}(x) = \left(1 + 8\,x + 16\,x^{2}\right)\,P_{n-2}(x)\)

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

\(P_{1}(x) = 2 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	4														6
2	1	8	16													25
3	2	20	64	64												150
4	1	16	96	256	256											625
5	2	36	256	896	1536	1024										3750
6	1	24	240	1280	3840	6144	4096									15625
7	2	52	576	3520	12800	27648	32768	16384								93750
8	1	32	448	3584	17920	57344	114688	131072	65536							390625
9	2	68	1024	8960	50176	186368	458752	720896	655360	262144						2343750
10	1	40	720	7680	53760	258048	860160	1966080	2949120	2621440	1048576					9765625
11	2	84	1600	18240	138240	731136	2752512	7372800	13762560	17039360	12582912	4194304				58593750
12	1	48	1056	14080	126720	811008	3784704	12976128	32440320	57671680	69206016	50331648	16777216			244140625
13	2	100	2304	32384	309760	2128896	10813440	41091072	116785152	245104640	369098752	377487360	234881024	67108864		1464843750
14	1	56	1456	23296	256256	2050048	12300288	56229888	196804608	524812288	1049624576	1526726656	1526726656	939524096	268435456	6103515625

Computed in your browser ?

Technical details

Node id: n-c322146fd9758ee0 · revision 1 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-2c2fe627ec960e5f

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: 2,  4;
  2: 1,  8,   16;
  3: 2, 20,   64,   64;
  4: 1, 16,   96,  256,   256;
  5: 2, 36,  256,  896,  1536,   1024;
  6: 1, 24,  240, 1280,  3840,   6144,   4096;
  7: 2, 52,  576, 3520, 12800,  27648,  32768,  16384;
  8: 1, 32,  448, 3584, 17920,  57344, 114688, 131072,  65536;
  9: 2, 68, 1024, 8960, 50176, 186368, 458752, 720896, 655360, 262144;

%F Formula

The row polynomials satisfy P(n,x) = (1+8*x+16*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 4*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 4*x;
P[n_, x_] := P[n, x] = Expand[(1 + 8*x + 16*x^2)*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (1 + 8*x + 16*x^2)*v[i - 2]); 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-d8ead6c583c2763e
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-d8ead6c583c2763e
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-d8ead6c583c2763e

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-d8ead6c583c2763e

Geometric basis map (elementary) polytool ?
- premise (sequence): Interpolation operator: Triangle with an even number of zeros in each line followed by 1 or 2 ones.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers periodic_product e-4d02ac33f5b2fbc9 root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

No candidate equalities or related sequences.