A299989 → 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

\(P_{n}(x) = \left(8 + 12\,x + 4\,x^{2}\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	x¹⁷	x¹⁸	x¹⁹	x²⁰	x²¹	x²²	x²³	x²⁴	x²⁵	x²⁶	x²⁷	x²⁸	x²⁹	P_n(1)
0	1	2																													3
1	8	28	28	8																											72
2	64	320	592	512	208	32																									1728
3	512	3328	8832	12480	10176	4800	1216	128																							41472
4	4096	32768	112640	219136	266496	210432	108032	34816	6400	512																					995328
5	32768	311296	1310720	3235840	5212160	5757952	4455424	2416640	901120	220160	31744	2048																			23887872
6	262144	2883584	14352384	42860544	85770240	121552896	125587456	95830016	54030336	22241280	6500352	1277952	151552	8192																	573308928
7	2097152	26214400	150470656	526647296	1257897984	2173108224	2806415360	2759901184	2084552704	1209614336	535019520	177192960	42549248	6995968	704512	32768															13759414272
8	16777216	234881024	1526726656	6123683840	16984834048	34586230784	53560213504	64448626688	61020897280	45731151872	27133739008	12676235264	4606787584	1275330560	259784704	36700160	3211264	131072													330225942528
9	134217728	2080374784	15099494400	68249714688	215469785088	505002590208	911455813632	1296656498688	1475791552512	1355894489088	1009927323648	609939357696	297504079872	116189036544	35809394688	8512339968	1505230848	186384384	14417920	524288											7925422620672
10	1073741824	18253611008	146297323520	735513149440	2603152834560	6898657001472	14213556731904	23330732113920	31012033658880	33743280537600	30253318668288	22422220701696	13739014225920	6939318681600	2870759915520	962567602176	257427505152	53603205120	8372879360	922746880	63963136	2097152									190210142896128
11	8589934592	158913789952	1393716887552	7712687521792	30236569763840	89369142624256	206904949211136	384803165700096	584919281565696	735413576663040	770994050433024	677390711783424	499992036900864	310071602970624	161193960407040	69906934530048	25093270929408	7368226111488	1739931516928	322269347840	45076185088	4475322368	281018368	8388608							4565043429507072
12	68719476736	1374389534720	13091060318208	79061757984768	340019675922432	1108642728247296	2848615584235520	5918761286631424	10124612037771264	14441552654893056	17332592449683456	17612708606115840	15212601038340096	11190040113709056	7010379066507264	3733869413007360	1684405223424000	639692798164992	202711249190912	52930237431808	11187567722496	1865794191360	236256755712	21340618752	1224736768	33554432					109561042308169728
13	549755813888	11819749998592	121496034869248	795084345835520	3721262744469504	13265624968986624	37452716116541440	85968048216866816	163416494078689280	260702810838925312	352457819607269376	406658988864700416	402383681378844672	342522367794216960	251213918049927168	158755664556982272	86323191009509376	40265882718437376	16035624465203200	5414748082405376	1535508387725312	360898115928064	69029855232000	10468982784000	1210912342016	100327751680	5301600256	134217728			2629465015396073472
14	4398046511104	101155069755392	1116004302192640	7865906185109504	39797098245259264	153960490068869120	473694279538049024	1190239479009378304	2488759395698081792	4390492608523141120	6601762263240015872	8525576989560537088	9508808595836239872	9195419074358673408	7729514483445399552	5654701804231852032	3600509174959570944	1993028012089540608	956768352380911616	396809009115430912	141403541951479808	42972277909749760	11025049783894016	2355702588768256	411434513072128	57209501253632	6089995190272	466003951616	22817013760	536870912	63107160369505763328

Computed in your browser ?

Technical details

Node id: n-bae5a2c240bb2e7e · revision 1 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-80a60edfe80ff887

Observed on the stored rows ?

Degrees: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29 (affine: slope 2, intercept 1)
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,          2;
  1:         8,         28,          28,           8;
  2:        64,        320,         592,         512,          208,           32;
  3:       512,       3328,        8832,       12480,        10176,         4800,         1216,           128;
  4:      4096,      32768,      112640,      219136,       266496,       210432,       108032,         34816,          6400,           512;
  5:     32768,     311296,     1310720,     3235840,      5212160,      5757952,      4455424,       2416640,        901120,        220160,         31744,         2048;
  6:    262144,    2883584,    14352384,    42860544,     85770240,    121552896,    125587456,      95830016,      54030336,      22241280,       6500352,      1277952,       151552,         8192;
  7:   2097152,   26214400,   150470656,   526647296,   1257897984,   2173108224,   2806415360,    2759901184,    2084552704,    1209614336,     535019520,    177192960,     42549248,      6995968,      704512,      32768;
  8:  16777216,  234881024,  1526726656,  6123683840,  16984834048,  34586230784,  53560213504,   64448626688,   61020897280,   45731151872,   27133739008,  12676235264,   4606787584,   1275330560,   259784704,   36700160,    3211264,    131072;
  9: 134217728, 2080374784, 15099494400, 68249714688, 215469785088, 505002590208, 911455813632, 1296656498688, 1475791552512, 1355894489088, 1009927323648, 609939357696, 297504079872, 116189036544, 35809394688, 8512339968, 1505230848, 186384384, 14417920, 524288;

%F Formula

The row polynomials satisfy P(n,x) = (8+12*x+4*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1 + 2*x.

%t Mathematica

P[0, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[(8 + 12*x + 4*x^2)*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 + 2*x; for(n = 1, N, my(i = n + 1); v[i] = (8 + 12*x + 4*x^2)*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-1cab7244184e4365
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-1cab7244184e4365
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-1cab7244184e4365

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-1cab7244184e4365

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle read by rows: T(n,0) = 0 for n >= 0; T(n,2*k+1) = A152842(2*n,2*(n-k)) and T(n,2*k) = A152842(2*n,2*(n-k)+1) f…
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers periodic_product e-0e7ef206328a472c root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

No candidate equalities or related sequences.