A299989 → 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

\(P_{n}(x) = \left(8 + 24\,x + 16\,x^{2}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1 + 4\,x\)

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	4																													5
1	8	56	112	64																											240
2	64	640	2368	4096	3328	1024																									11520
3	512	6656	35328	99840	162816	153600	77824	16384																							552960
4	4096	65536	450560	1753088	4263936	6733824	6914048	4456448	1638400	262144																					26542080
5	32768	622592	5242880	25886720	83394560	184254464	285147136	309329920	230686720	112721920	32505856	4194304																			1274019840
6	262144	5767168	57409536	342884352	1372323840	3889692672	8037597184	12266242048	13831766016	11387535360	6656360448	2617245696	620756992	67108864																	61152952320
7	2097152	52428800	601882624	4213178368	20126367744	69539463168	179610583040	353267351552	533645492224	619322540032	547859988480	362891182080	174281719808	57310969856	11542724608	1073741824															2935341711360
8	16777216	469762048	6106906624	48989470720	271757344768	1106759385088	3427853664256	8249424216064	15621349703680	23414349758464	27784948744192	25960929820672	18869401944064	10447507947520	4256312590336	1202590842880	210453397504	17179869184													140896402145280
9	134217728	4160749568	60397977600	545997717504	3447516561408	16160082886656	58333172072448	165972031832064	377802637443072	694217978413056	1034165579415552	1249155804561408	1218576711155712	951820587368448	586701122568192	278932356071424	98646808854528	24429773979648	3779571220480	274877906944											6763027302973440
10	1073741824	36507222016	585189294080	5884105195520	41650445352960	220757024047104	909667630841856	2986333710581760	7939080616673280	17276559635251200	30979398316326912	45920707997073408	56275002269368320	56846898639667200	47034530455879680	31541415188103168	16870768977641472	7025879301488640	2194900086947840	483785116221440	67070209294336	4398046511104									324625310542725120
11	8589934592	317827579904	5574867550208	61701500174336	483785116221440	2859812563976192	13241916749512704	49254805209612288	149739336080818176	376531751251476480	789497907643416576	1387296177732452352	2047967383145938944	2540106571535351808	2641001847308943360	2290710430680612864	1644512603629682688	965768132884955136	456112607573573632	168961951840337920	47265805854834688	9385431254695936	1178676464975872	70368744177664							15582014906050805760
12	68719476736	2748779069440	52364241272832	632494063878144	5440314814758912	35476567303913472	182311397391073280	757601444688822272	2591900681669443584	7394074959305244672	17748574668475858944	36070827225325240320	62310813853041033216	91668808611504586752	114858050625655013376	122351432925425172480	110389180722315264000	83845814441081831424	53139537707902435328	27750688322647752704	11731015012183965696	3912854019999006720	990932655509864448	179018085187977216	20547673299877888	1125899906842624					747936715490438676480
13	549755813888	23639499997184	485984139476992	6360674766684160	59540203911512064	424499999007571968	2396973831458652160	11003910171758952448	41834622484144455680	133479839149529759744	360916807277843841024	832837609194906451968	1648163558927747776512	2805943236970225336320	4115888833330006720512	5202105616203195088896	5657276645999206465536	5277729779671023747072	4203642739806227660800	2838887442628149772288	1610097243167456755712	756858205614771273728	289532197918998528000	87820192733724672000	20315737919068307456	3366440721459445760	355784370562269184	18014398509481984			35900962343541056471040
14	4398046511104	202310139510784	4464017208770560	62927249480876032	636753571924148224	4926735682203811840	30316433890435137536	152350653313200422912	637122405298708938752	2247932215563848253440	6760204557557776252928	17460381674619979956224	38948080008545238515712	75328873057146252558336	126640365296769426259968	185293268721069327384576	235962969290150441385984	261230167600600266571776	250811082966541694664704	208042201771111041990656	148272360405314891153408	90119398562987528683520	46242410408785806884864	19761065581762102427648	6902725695665915035648	1919632319568909697024	408692658984617771008	62545991624921448448	6124895493223874560	288230376151711744	1723246192489970710609920

Computed in your browser ?

Technical details

Node id: n-f2d3e5cb5c029ac6 · revision 1 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-d3a321e8b3ce7458

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,          4;
  1:         8,         56,         112,           64;
  2:        64,        640,        2368,         4096,          3328,           1024;
  3:       512,       6656,       35328,        99840,        162816,         153600,          77824,           16384;
  4:      4096,      65536,      450560,      1753088,       4263936,        6733824,        6914048,         4456448,         1638400,          262144;
  5:     32768,     622592,     5242880,     25886720,      83394560,      184254464,      285147136,       309329920,       230686720,       112721920,         32505856,          4194304;
  6:    262144,    5767168,    57409536,    342884352,    1372323840,     3889692672,     8037597184,     12266242048,     13831766016,     11387535360,       6656360448,       2617245696,        620756992,        67108864;
  7:   2097152,   52428800,   601882624,   4213178368,   20126367744,    69539463168,   179610583040,    353267351552,    533645492224,    619322540032,     547859988480,     362891182080,     174281719808,     57310969856,     11542724608,      1073741824;
  8:  16777216,  469762048,  6106906624,  48989470720,  271757344768,  1106759385088,  3427853664256,   8249424216064,  15621349703680,  23414349758464,   27784948744192,   25960929820672,   18869401944064,  10447507947520,   4256312590336,   1202590842880,   210453397504,    17179869184;
  9: 134217728, 4160749568, 60397977600, 545997717504, 3447516561408, 16160082886656, 58333172072448, 165972031832064, 377802637443072, 694217978413056, 1034165579415552, 1249155804561408, 1218576711155712, 951820587368448, 586701122568192, 278932356071424, 98646808854528, 24429773979648, 3779571220480, 274877906944;

%F Formula

The row polynomials satisfy P(n,x) = (8+24*x+16*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1 + 4*x.

%t Mathematica

P[0, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[(8 + 24*x + 16*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 + 4*x; for(n = 1, N, my(i = n + 1); v[i] = (8 + 24*x + 16*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-6938cf52b8fdd4be
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-6938cf52b8fdd4be
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-6938cf52b8fdd4be

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-6938cf52b8fdd4be

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.