A122431 → 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

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

\(P_{0}(x) = 8 + 48\,x + 96\,x^{2} + 64\,x^{3}\)

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	48	96	64															216
1	8	80	288	448	256														1080
2	8	112	608	1600	2048	1024													5400
3	8	144	1056	4032	8448	9216	4096												27000
4	8	176	1632	8256	24576	43008	40960	16384											135000
5	8	208	2336	14784	57600	141312	212992	180224	65536										675000
6	8	240	3168	24128	116736	371712	778240	1032192	786432	262144									3375000
7	8	272	4128	36800	213248	838656	2265088	4145152	4915200	3407872	1048576								16875000
8	8	304	5216	53312	360448	1691648	5619712	13205504	21495808	23068672	14680064	4194304							84375000
9	8	336	6432	74176	573696	3133440	12386304	35684352	74317824	109051904	106954752	62914560	16777216						421875000
10	8	368	7776	99904	870400	5428224	24920064	85229568	217055232	406323200	543162368	490733568	268435456	67108864					2109375000
11	8	400	9248	131008	1270016	8909824	46632960	184909824	557973504	1274544128	2168455168	2663383040	2231369728	1140850688	268435456				10546875000
12	8	432	10848	168000	1794048	13989888	82272256	371441664	1297612800	3506438144	7266631680	11337203712	12884901888	10066329600	4831838208	1073741824			52734375000
13	8	464	12576	211392	2466048	21166080	138231808	700530688	2783379456	8696889344	21292384256	40403730432	58233716736	61605937152	45097156608	20401094656	4294967296		263671875000
14	8	496	14432	261696	3311616	31030272	222896128	1253457920	5585502208	19830407168	56079941632	125573267456	219848638464	294540804096	291520905216	200789721088	85899345920	17179869184	1318359375000

Computed in your browser ?

Technical details

Node id: n-c77d53cf93e5da02 · revision 1 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-751aae81c1f2029f

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,  48,   96,    64;
  1: 8,  80,  288,   448,    256;
  2: 8, 112,  608,  1600,   2048,    1024;
  3: 8, 144, 1056,  4032,   8448,    9216,     4096;
  4: 8, 176, 1632,  8256,  24576,   43008,    40960,    16384;
  5: 8, 208, 2336, 14784,  57600,  141312,   212992,   180224,    65536;
  6: 8, 240, 3168, 24128, 116736,  371712,   778240,  1032192,   786432,    262144;
  7: 8, 272, 4128, 36800, 213248,  838656,  2265088,  4145152,  4915200,   3407872,   1048576;
  8: 8, 304, 5216, 53312, 360448, 1691648,  5619712, 13205504, 21495808,  23068672,  14680064,  4194304;
  9: 8, 336, 6432, 74176, 573696, 3133440, 12386304, 35684352, 74317824, 109051904, 106954752, 62914560, 16777216;

%F Formula

The row polynomials satisfy P(n,x) = (1+4*x)*P(n-1,x) for n >= 1, with P(0,x) = 8 + 48*x + 96*x^2 + 64*x^3.

%t Mathematica

P[0, x_] := 8 + 48*x + 96*x^2 + 64*x^3;
P[n_, x_] := P[n, x] = Expand[(1 + 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] = 8 + 48*x + 96*x^2 + 64*x^3; for(n = 1, N, my(i = n + 1); v[i] = (1 + 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-4464410b9667d6e5
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-4464410b9667d6e5
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-4464410b9667d6e5

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-4464410b9667d6e5

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.