A159854 → 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

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

\(P_{0}(x) = 1 + 8\,x + 16\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	1	8	16															25
1	2	20	64	64														150
2	4	48	208	384	256													900
3	8	112	608	1600	2048	1024												5400
4	16	256	1664	5632	10496	10240	4096											32400
5	32	576	4352	17920	43520	62464	49152	16384										194400
6	64	1280	11008	53248	158720	299008	348160	229376	65536									1166400
7	128	2816	27136	150528	530432	1232896	1892352	1851392	1048576	262144								6998400
8	256	6144	65536	409600	1662976	4587520	8716288	11272192	9502720	4718592	1048576							41990400
9	512	13312	155648	1081344	4964352	15826944	35782656	57409536	64094208	47448064	20971520	4194304						251942400
10	1024	28672	364544	2785280	14254080	51511296	134873088	257949696	357826560	351272960	231735296	92274688	16777216					1511654400
11	2048	61440	843776	7028736	39649280	160038912	475791360	1055391744	1747451904	2133852160	1868562432	1111490560	402653184	67108864				9069926400
12	4096	131072	1933312	17432576	107413504	478674944	1591738368	4013948928	7716470784	11257511936	12272533504	9697230848	5251268608	1744830464	268435456			54419558400
13	8192	278528	4390912	42598400	284557312	1387003904	5098176512	14394851328	31488737280	53380907008	69575114752	68484595712	49291460608	24494735360	7516192768	1073741824		326517350400
14	16384	589824	9895936	102760448	739508224	3912237056	15744368640	49182408704	120556879872	232716763136	352673857536	415269650432	372521304064	246155313152	113011326976	32212254720	4294967296	1959104102400

Computed in your browser ?

Technical details

Node id: n-84c761192e0db3c0 · revision 1 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-23ef07238c102b65

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 ?

A055846: 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:   1,     8,     16;
  1:   2,    20,     64,      64;
  2:   4,    48,    208,     384,     256;
  3:   8,   112,    608,    1600,    2048,     1024;
  4:  16,   256,   1664,    5632,   10496,    10240,     4096;
  5:  32,   576,   4352,   17920,   43520,    62464,    49152,    16384;
  6:  64,  1280,  11008,   53248,  158720,   299008,   348160,   229376,    65536;
  7: 128,  2816,  27136,  150528,  530432,  1232896,  1892352,  1851392,  1048576,   262144;
  8: 256,  6144,  65536,  409600, 1662976,  4587520,  8716288, 11272192,  9502720,  4718592,  1048576;
  9: 512, 13312, 155648, 1081344, 4964352, 15826944, 35782656, 57409536, 64094208, 47448064, 20971520, 4194304;

%F Formula

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

%t Mathematica

P[0, x_] := 1 + 8*x + 16*x^2;
P[n_, x_] := P[n, x] = Expand[(2 + 4*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] = 1 + 8*x + 16*x^2; for(n = 1, N, my(i = n + 1); v[i] = (2 + 4*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-d3bf3843e93b142d
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-d3bf3843e93b142d
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-d3bf3843e93b142d

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-d3bf3843e93b142d

Geometric basis map (elementary) polytool ?
- premise (sequence): Riordan array (1-x,x/(1-x)).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers periodic_product e-b62074ba45057e82 root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

No candidate equalities or related sequences.