Real-rooted atlas

A038319 → x ↦ x+1 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(16 + 37\,x + 21\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 256 + 672\,x + 441\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
02566724411369
140961612821168926150653
2655363440646773765927041944811874161
310485766881280180633602370816015558480408410169343957
416777216132120576433520640758661120746807040392073696857661212565726409
5268435456246625075297108623362124251136027880796160219561269769605805552180108854194931877133
6429496729645097156608207165063168543808290816892185477120936794750976614771555328230539333248378228593613512479453921
768719476736811748818944426168129945613051398979584256949417410563372461103513629509034655744165988319938565446491747984794280046581129961739795077
81099511627776144310901145608523362598912029831769096192068519844642816010791875531243521180361386229760885271039672320435719339838720127084807452960166798809782014808584372417849
917592186044416253987186016256166679090823168065629892011622401722784665305088031656168224980992415487207952875523895192574558208025562201270538240111834630558604802935659052163376350277500542221177917621779460413
102814749767106564433230883192832320023854380482561400104362914611204134683196732211208682834713137643521329559065449201664149575394863035187212269856609858355207157416355750707202818232690076840966725328010410643273558275113866416582952005840035281
114503599627370496768426686420090886051360155558215682912217074862391296955571227689222144022575370254157873152395068979447762780165185280355251886489651042603497010757632372185650499036774401953974665119943065669943411308270689281530012122368421328154472377739119461243569224216081305397
127205759403792793613240582904469258241129587229037533593659303329524470513664214047955002385760256561875881881262620672110619314245373578444816592897136806036766721905590530555068284928166739171423568474931210942258124671681167365222441377684211466241713613577052631887363460181261356275926432439199325215086819012061295995008299689
13115292150460684697622698142121947299840208539180745390817280118606659048941027328046701372000520529510401348502116515030289612829498483798766287585280497786914104181103001606533453247617376976896066695668569427389972480525228389984240696033283133464826610526879744013708908616421055098880415221751362753111168077854078380516208344068122318582951682301333446267951815307088493
141844674407370955161638738162554790058393638132878764871463731202335638824348377153536099629593601110462955520313833219843497958309888755161185248416962183168141592722234078180409344020907050392375606326067202439155879110487404707840224097446393276030307532816043339912245897624289288773701514509475263283203543225611628826548633609965322032706074668032017439313557235630669056143056869024198532832112337511914217166362274241
Technical details
Node id
n-b0fdf8a45457ba81 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-039ab5acdf7230f1

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,             672,              441;
  1:           4096,           16128,            21168,             9261;
  2:          65536,          344064,           677376,           592704,            194481;
  3:        1048576,         6881280,         18063360,         23708160,          15558480,           4084101;
  4:       16777216,       132120576,        433520640,        758661120,         746807040,         392073696,          85766121;
  5:      268435456,      2466250752,       9710862336,      21242511360,       27880796160,       21956126976,        9605805552,        1801088541;
  6:     4294967296,     45097156608,     207165063168,     543808290816,      892185477120,      936794750976,      614771555328,      230539333248,       37822859361;
  7:    68719476736,    811748818944,    4261681299456,   13051398979584,    25694941741056,    33724611035136,    29509034655744,    16598831993856,     5446491747984,      794280046581;
  8:  1099511627776,  14431090114560,   85233625989120,  298317690961920,   685198446428160,  1079187553124352,  1180361386229760,   885271039672320,   435719339838720,   127084807452960,   16679880978201;
  9: 17592186044416, 253987186016256, 1666790908231680, 6562989201162240, 17227846653050880, 31656168224980992, 41548720795287552, 38951925745582080, 25562201270538240, 11183463055860480, 2935659052163376, 350277500542221;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (16+37*x+21*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 256 + 672*x + 441*x^2.
%t Mathematica
P[0, x_] := 256 + 672*x + 441*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((16 + 37*x + 21*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 + 672*x + 441*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((16 + 37*x + 21*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

Same fingerprint ?

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.