A139548 → 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

\(P_{n}(x) = \left(8 + 32\,x + 32\,x^{2}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1\)

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	P_n(1)
0	1																													1
1	8	32	32																											72
2	64	512	1536	2048	1024																									5184
3	512	6144	30720	81920	122880	98304	32768																							373248
4	4096	65536	458752	1835008	4587520	7340032	7340032	4194304	1048576																					26873856
5	32768	655360	5898240	31457280	110100480	264241152	440401920	503316480	377487360	167772160	33554432																			1934917632
6	262144	6291456	69206016	461373440	2076180480	6643777536	15502147584	26575110144	33218887680	29527900160	17716740096	6442450944	1073741824																	139314069504
7	2097152	58720256	763363328	6106906624	33587986432	134351945728	403055837184	921270484992	1612223348736	2149631131648	2149631131648	1563368095744	781684047872	240518168576	34359738368															10030613004288
8	16777216	536870912	8053063680	75161927680	488552529920	2345052143616	8598524526592	24567212933120	55276229099520	98268851732480	137576392425472	150083337191424	125069447659520	76965813944320	32985348833280	8796093022208	1099511627776													722204136308736
9	134217728	4831838208	82141249536	876173328384	6571299962880	36799279792128	159463545765888	546732156911616	1503513431506944	3341140958904320	6014053726027776	8747714510585856	10205666929016832	9420615626784768	6729011161989120	3588805953060864	1345802232397824	316659348799488	35184372088832											51998697814228992
10	1073741824	42949672960	816043786240	9792525434880	83236466196480	532713383657472	2663566918287360	10654267673149440	34626369937735680	92336986500628480	203141370301382656	369347946002513920	554021919003770880	681873131081564160	681873131081564160	545498504865251328	340936565540782080	160440736725073920	53480245575024640	11258999068426240	1125899906842624									3743906242624487424
11	8589934592	377957122048	7937099563008	105827994173440	1005365944647680	7238634801463296	41018930541625344	187515111047430144	703181666427863040	2187676295553351680	5687958368438714368	12410090985684467712	22751833473754857472	35002820728853626880	45003626651383234560	48003868428142116864	42003384874624352256	29649448146793660416	16471915637107589120	6935543426150563840	2080663027845169152	396316767208603648	36028797018963968							269561249468963094528
12	68719476736	3298534883328	75866302316544	1112705767309312	11683410556747776	93467284453982208	591959468208553984	3044362979358277632	12938542662272679936	46003707243636195328	138011121730908585984	351301037133221855232	761152247121980686336	1405204148532887420928	2208177947694537375744	2944237263592716500992	3312266921541806063616	3117427690862876295168	2424665981782237118464	1531367988494044495872	765683994247022247936	291689140665532284928	79551583817872441344	13835058055282163712	1152921504606846976					19408409961765342806016
13	549755813888	28587302322176	714682558054400	11434920928870400	131501590682009600	1157213998001684480	8100497986011791360	46288559920067379200	219870659620320051200	879482638481280204800	2990240970836352696320	8698882824251207843840	21747207060628019609600	46840138284429580697600	86988828242512078438400	139182125188019325501440	191375422133526572564480	225147555451207732428800	225147555451207732428800	189597941432595985203200	132718559002817189642240	75839176573038394081280	34472352987744724582400	11990383647911208550400	2997595911977802137600	479615345916448342016	36893488147419103232			1397405517247104682033152
14	4398046511104	246290604621824	6649846324789248	115264002963013632	1440800037037670400	13831680355561635840	106042882725972541440	666555262848970260480	3499415129957093867520	15552956133142639411200	59101233305942029762560	193422218092173915586560	548029617927826094161920	1348995982591571924090880	2890705676981939837337600	5395983930366287696363520	8768473886845217506590720	12379021957899130597539840	15129915726321159619215360	15926227080338062757068800	14333604372304256481361920	10920841426517528747704320	6949626362329336475811840	3625892015128349465640960	1510788339636812277350400	483452268683779928752128	111565908157795368173568	16528282690043758247936	1180591620717411303424	100613197241791537106386944

Computed in your browser ?

Technical details

Node id: n-f8ceb867f43c1485 · revision 1 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-90334921cba1a1bb

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, intercept 0)
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;
  1:         8,         32,          32;
  2:        64,        512,        1536,         2048,          1024;
  3:       512,       6144,       30720,        81920,        122880,          98304,           32768;
  4:      4096,      65536,      458752,      1835008,       4587520,        7340032,         7340032,         4194304,          1048576;
  5:     32768,     655360,     5898240,     31457280,     110100480,      264241152,       440401920,       503316480,        377487360,        167772160,         33554432;
  6:    262144,    6291456,    69206016,    461373440,    2076180480,     6643777536,     15502147584,     26575110144,      33218887680,      29527900160,      17716740096,       6442450944,        1073741824;
  7:   2097152,   58720256,   763363328,   6106906624,   33587986432,   134351945728,    403055837184,    921270484992,    1612223348736,    2149631131648,    2149631131648,    1563368095744,      781684047872,     240518168576,      34359738368;
  8:  16777216,  536870912,  8053063680,  75161927680,  488552529920,  2345052143616,   8598524526592,  24567212933120,   55276229099520,   98268851732480,  137576392425472,  150083337191424,   125069447659520,   76965813944320,   32985348833280,    8796093022208,    1099511627776;
  9: 134217728, 4831838208, 82141249536, 876173328384, 6571299962880, 36799279792128, 159463545765888, 546732156911616, 1503513431506944, 3341140958904320, 6014053726027776, 8747714510585856, 10205666929016832, 9420615626784768, 6729011161989120, 3588805953060864, 1345802232397824, 316659348799488, 35184372088832;

%F Formula

The row polynomials satisfy P(n,x) = (8+32*x+32*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(8 + 32*x + 32*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; for(n = 1, N, my(i = n + 1); v[i] = (8 + 32*x + 32*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-770b6439e58a100f
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-770b6439e58a100f
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-770b6439e58a100f

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-770b6439e58a100f

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers periodic_product e-4755dd2fd7289161 root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (5)

Equal up to content, sign, reversal and a small index shift.