A122431 → 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

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

\(P_{0}(x) = 8 + 24\,x + 24\,x^{2} + 8\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	x¹⁷	P_n(1)
0	8	24	24	8															64
1	8	40	72	56	16														192
2	8	56	152	200	128	32													576
3	8	72	264	504	528	288	64												1728
4	8	88	408	1032	1536	1344	640	128											5184
5	8	104	584	1848	3600	4416	3328	1408	256										15552
6	8	120	792	3016	7296	11616	12160	8064	3072	512									46656
7	8	136	1032	4600	13328	26208	35392	32384	19200	6656	1024								139968
8	8	152	1304	6664	22528	52864	87808	103168	83968	45056	14336	2048							419904
9	8	168	1608	9272	35856	97920	193536	278784	290304	212992	104448	30720	4096						1259712
10	8	184	1944	12488	54400	169632	389376	665856	847872	793600	530432	239616	65536	8192					3779136
11	8	200	2312	16376	79376	278432	728640	1444608	2179584	2489344	2117632	1300480	544768	139264	16384				11337408
12	8	216	2712	21000	112128	437184	1285504	2901888	5068800	6848512	7096320	5535744	3145728	1228800	294912	32768			34012224
13	8	232	3144	26424	154128	661440	2159872	5472896	10872576	16986112	20793344	19728384	14217216	7520256	2752512	622592	65536		102036672
14	8	248	3608	32712	206976	969696	3482752	9792640	21818368	38731264	54765568	61315072	53673984	35954688	17793024	6127616	1310720	131072	306110016

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Technical details

Node id: n-48e70e0cceeb235f · revision 1 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-f9d50fb492dc5697

Observed on the stored rows ?

Degrees: 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 (affine: slope 1, intercept 3)
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: 8,  24,   24,    8;
  1: 8,  40,   72,   56,    16;
  2: 8,  56,  152,  200,   128,    32;
  3: 8,  72,  264,  504,   528,   288,     64;
  4: 8,  88,  408, 1032,  1536,  1344,    640,    128;
  5: 8, 104,  584, 1848,  3600,  4416,   3328,   1408,    256;
  6: 8, 120,  792, 3016,  7296, 11616,  12160,   8064,   3072,    512;
  7: 8, 136, 1032, 4600, 13328, 26208,  35392,  32384,  19200,   6656,   1024;
  8: 8, 152, 1304, 6664, 22528, 52864,  87808, 103168,  83968,  45056,  14336,  2048;
  9: 8, 168, 1608, 9272, 35856, 97920, 193536, 278784, 290304, 212992, 104448, 30720, 4096;

%F Formula

The row polynomials satisfy P(n,x) = (1+2*x)*P(n-1,x) for n >= 1, with P(0,x) = 8 + 24*x + 24*x^2 + 8*x^3.

%t Mathematica

P[0, x_] := 8 + 24*x + 24*x^2 + 8*x^3;
P[n_, x_] := P[n, x] = Expand[(1 + 2*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] = 8 + 24*x + 24*x^2 + 8*x^3; for(n = 1, N, my(i = n + 1); v[i] = (1 + 2*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-5fbd3e16dd967779
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-5fbd3e16dd967779
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-5fbd3e16dd967779

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-5fbd3e16dd967779

Geometric basis map (elementary) polytool ?
- premise (sequence): Riordan array ((1+x)^3,x).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers periodic_product e-73f8c6c9e8d957af root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

No candidate equalities or related sequences.