Real-rooted atlas

A038319 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(16 + 47\,x + 31\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 256 + 992\,x + 961\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
02569929612209
14096238084612829791103823
265536507904147609619066249235214879681
310485761015808039362560762649607388168028629151229345007
41677721619503513694470144024404787203546320640274839849688750368110779215329
5268435456364065587221161312256683334041601323959705601539103157769940041227227512614111506623120463
64294967296665719930884514413281281749335146496423667105792065668401397766361626385408352161460620885289103744123811286661761
7687194767361198295875584928679303577641984043515904122016126468096236406245031936305358066499584253556251646976122816309391504264396221606711119130473102767
81099511627776213030377881601857358607155209596352803635203253763372482560756499984102195212214322659983360135230000878387209825304751320320423033954570736081962828698080152599132235830049
91759218604441637493346507161636321679428812802111197616799744081808907650990080221906662003310592429944157631414272595012003864903680576417878744125440372269880022247680144254578508620976254084768964048312472159215084012303
102814749767106566544293208522752697376245033205764503888249172787201963413783623761920608658272923366195213758213044205256704228484609484123013122766805817971802112023825272321423851520138484395368276136964878427564109727552787662783788549761116191483108948578241
114503599627370496113434415614394368131867508151733452893680875582793973764537667411041583104015825115096007521075240881547331352762777679207997954495977881611509912202762696785921238914160714040279040960158474553381216256507356466667411665408163833859028018350288244175462974450425915460999706120583177327
127205759403792793619545622382787952642461526818832357785619076832845950772838410164375000733146152963938695312784094134272114468332527787735777282534655934543871292211242970338890314068000768555033543999890045009925376887457498934811033637882616177833404350464183493922111380552322565469530370627689540384756943935220796320321256666986187667409334369
1311529215046068469763350678122763649024045443572039981989888038153665691901545676802217681818341777342464094528687506818259222528305248886740767295406080760396780363161387663360147326876195362518859776022201341759995601800396802580905979599488709296128227295697067000426102784014679513768910444185804806563436444753227448460801816665444529911168770402346526199184468592995112063348350820368238715343
1418446744073709551616571849066284996100096830968174445384957952075133372439436889948160473105454579579166392320219994036379504312372428878143715005636427623956482162906397477436836020224047144600382516006035128320811934784365553437271654401101186551295781849299681281163753968983042181646254089394888812102684278915072056007990995227540893532160232533176899828629602611206007107069912239598067456727423121747185263828481566977372488557307219621121
Technical details
Node id
n-83a7464f2b086513 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-4dc50714404e5462

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:            256,             992,              961;
  1:           4096,           23808,            46128,             29791;
  2:          65536,          507904,          1476096,           1906624,            923521;
  3:        1048576,        10158080,         39362560,          76264960,          73881680,           28629151;
  4:       16777216,       195035136,        944701440,        2440478720,        3546320640,         2748398496,          887503681;
  5:      268435456,      3640655872,      21161312256,       68333404160,      132395970560,       153910315776,        99400412272,        27512614111;
  6:     4294967296,     66571993088,     451441328128,     1749335146496,     4236671057920,      6566840139776,      6361626385408,      3521614606208,       852891037441;
  7:    68719476736,   1198295875584,    9286793035776,    41984043515904,   122016126468096,    236406245031936,    305358066499584,    253556251646976,    122816309391504,     26439622160671;
  8:  1099511627776,  21303037788160,  185735860715520,   959635280363520,  3253763372482560,   7564999841021952,  12214322659983360,  13523000087838720,   9825304751320320,   4230339545707360,    819628286980801;
  9: 17592186044416, 374933465071616, 3632167942881280, 21111976167997440, 81808907650990080, 221906662003310592, 429944157631414272, 595012003864903680, 576417878744125440, 372269880022247680, 144254578508620976, 25408476896404831;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (16+47*x+31*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 256 + 992*x + 961*x^2.
%t Mathematica
P[0, x_] := 256 + 992*x + 961*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((16 + 47*x + 31*x^2)*P[n - 1, x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 256 + 992*x + 961*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((16 + 47*x + 31*x^2)*v[i - 1])/(1 + x)); 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)

polytool c48629374597

Properties

Proof chains ? JSON

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.