Real-rooted atlas

A038226 → basis A013611 → basis A007318

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(43 + 32\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1849 + 2752\,x + 1024\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
01849275210245625
17950717750413209632768421875
2341880110176896113602565636096104857631640625
3147008443547008160814151680605880320225443840335544322373046875
463213630492822562105652512783360521057075202908225536086570434561073741824177978515625
5271818611107141598532297631612695582723920954490880291791962112013028850401283231962890243435973836813348388671875
61168820027760169585564443392181246121340928269761668972544250941087416320149397484601344555897617121281181974999859210995116277761001129150390625
7502592611936843336620167994908810020321279848448173996276487290881942284016602316814454206635180032717107926086451222871216247275524255109999493123518437208883275084686279296875
821611482313284249160829635819789760538592268791854080106883426985050112013919702118983270401243061770625482752770891020542935040327820766210949120914848649891020801512927999819776011258999068426245631351470947265625
9929293739471222707760724177427605564828306015904282997760631948262049108787209405741574684409856097994702917642223616729262905433616547843876480560444473344014424113713281761280357807471957377024053255065593656115236028797018963968422351360321044921875
1039959630797262576401356848795956949519488146059042066100268441636231700357482237132806066703315671444357120722360952935762676940862716609867291023114244000527938378696491008186071066901334720512061542885176668848128013739806923163277721618590859261785407488115292150460684697631676352024078369140625
111718264124282290785243166232064116612317828167422454955904550005350420253520499832570557235237680968371781526618112050474971586386417051238450083692736879545601228837271585292561522307891220802745279569221753241686006182034394715258880256018402334942408212485196145163669021392896644483121075227459584368934881474191032322375726401805877685546875
127388535734413850376544976978232767846627178886437235982362121159193507841108419940081746134132326422683942959812479024103424337621476611162478498283523768797878450185806492467232053563351602909184786432208720877638344524924190721035514431694112371716915238530769551408832435978241042693129509583626174464193989419443643465334784222098798647463001456641180591620717411303424178179480135440826416015625
133177070365797955661914307354649715251864818074155201847477586428319052293273605957757177939385470961254401330103928098095360958791680217765852414199798631392870427009718128892998279862681602584318545222984553023406080192321380109617455108718592011131780140711707995956838404970469272131739384241192961681342671334203597206323204170772518038334504697856071626862563806817969766407614815953627302907084803777893186295716170956813363461010158061981201171875
141366140257293120934623152011626660027288553298900125184907903271044773934269837312031530284141709978492471869440762592918776241340283040563201362026422372449649621802680321858268607267838281654552494081975567954570459302755670425601653963868942710113934979891201094094962401379300174043545605699471432044394493929901260823135275157558641497559007232717372873102593534808031232016426427147966363587733094402619496688047792200037171202599190512171452725618278412089258196146291747061761002259575761854648590087890625
Technical details
Node id
n-31033f5999f9a349 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-5894cfd190cff20e

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:               1849,                2752,                 1024;
  1:              79507,              177504,               132096,                32768;
  2:            3418801,            10176896,             11360256,              5636096,              1048576;
  3:          147008443,           547008160,            814151680,            605880320,            225443840,             33554432;
  4:         6321363049,         28225621056,          52512783360,          52105707520,          29082255360,           8657043456,           1073741824;
  5:       271818611107,       1415985322976,        3161269558272,        3920954490880,        2917919621120,        1302885040128,         323196289024,          34359738368;
  6:     11688200277601,      69585564443392,      181246121340928,      269761668972544,      250941087416320,      149397484601344,       55589761712128,       11819749998592,        1099511627776;
  7:    502592611936843,    3366201679949088,    10020321279848448,    17399627648729088,    19422840166023168,    14454206635180032,     7171079260864512,     2287121624727552,      425510999949312,      35184372088832;
  8:  21611482313284249,  160829635819789760,   538592268791854080,  1068834269850501120,  1391970211898327040,  1243061770625482752,   770891020542935040,   327820766210949120,    91484864989102080,   15129279998197760,   1125899906842624;
  9: 929293739471222707, 7607241774276055648, 28306015904282997760, 63194826204910878720, 94057415746844098560, 97994702917642223616, 72926290543361654784, 38764805604444733440, 14424113713281761280, 3578074719573770240, 532550655936561152, 36028797018963968;
%F Formula
The row polynomials satisfy P(n,x) = (43+32*x)*P(n-1,x) for n >= 1, with P(0,x) = 1849 + 2752*x + 1024*x^2.
%t Mathematica
P[0, x_] := 1849 + 2752*x + 1024*x^2;
P[n_, x_] := P[n, x] = Expand[(43 + 32*x)*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] = 1849 + 2752*x + 1024*x^2; for(n = 1, N, my(i = n + 1); v[i] = (43 + 32*x)*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)

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.