A133080 → 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

\(P_{n}(x) = \left(1 + 4\,x + 4\,x^{2}\right)\,P_{n-2}(x)\)

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

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

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	2														4
2	1	4	4													9
3	2	10	16	8												36
4	1	8	24	32	16											81
5	2	18	64	112	96	32										324
6	1	12	60	160	240	192	64									729
7	2	26	144	440	800	864	512	128								2916
8	1	16	112	448	1120	1792	1792	1024	256							6561
9	2	34	256	1120	3136	5824	7168	5632	2560	512						26244
10	1	20	180	960	3360	8064	13440	15360	11520	5120	1024					59049
11	2	42	400	2280	8640	22848	43008	57600	53760	33280	12288	2048				236196
12	1	24	264	1760	7920	25344	59136	101376	126720	112640	67584	24576	4096			531441
13	2	50	576	4048	19360	66528	168960	321024	456192	478720	360448	184320	57344	8192		2125764
14	1	28	364	2912	16016	64064	192192	439296	768768	1025024	1025024	745472	372736	114688	16384	4782969

Generated b-files: A133125 (row sums) ?

Computed in your browser ?

Technical details

Node id: n-f1bdd4180f3e6ae5 · revision 1 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-7e6e1abd41f83340

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 ?

A133125: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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,  2;
  2: 1,  4,   4;
  3: 2, 10,  16,    8;
  4: 1,  8,  24,   32,   16;
  5: 2, 18,  64,  112,   96,   32;
  6: 1, 12,  60,  160,  240,  192,   64;
  7: 2, 26, 144,  440,  800,  864,  512,  128;
  8: 1, 16, 112,  448, 1120, 1792, 1792, 1024,  256;
  9: 2, 34, 256, 1120, 3136, 5824, 7168, 5632, 2560, 512;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[(1 + 4*x + 4*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 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (1 + 4*x + 4*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-d9920fdddfeeaf81
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-d9920fdddfeeaf81
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-d9920fdddfeeaf81

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-d9920fdddfeeaf81

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.