Real-rooted atlas

A272867 → basis A013610 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(33 + 45\,x + 9\,x^{2}\right)\,P_{n-1}(x) + \left(45\,x + 18\,x^{2}\right)\,P_{n-1}'(x) + \left(9\,x^{2}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27x28Pn(1)
011
13345987
210894995440111348111700
335937438615945027653184177876196837292271051
41185921358291351612049852255015701302686553531130246758062886846561594272619
5391353932848038975249978116796344141454065591738550327372970418618680503124278462097332435383818559049199618130970
612914679692239081327353704470120701161344491246542756062841912122584142245168994836467570425155554244483773914887423374916070517468465674782969053144183085745467645
7426184429771752295041193553644812259286738839249586445841047761241696489613261657870542627899731781804816135516503137384728582154250635181306185787179001264127227574476938839334028280101374569173311478296941728450587894675
81406408618241136870796206489576743514727532945905841345302827290375265018985488123570694421247103647147040085434017534962407444267477038294814581273002936878073360193446482905505110714967271395451877850515936203505340885829301545647043351642432587780465431015924304672124771383398891535724
9464114844019531068225094288826551090707683586979779920732532172181641098012949696957609445746503543628294943886294837502211749223664070073140988034324103741518602068030654655331261645677173052116483979355148760694571658476635223983404865143697602337886644586214976169432445801056662844353589302433828213199754544860054049999897322247242138742048917096736975497726037183
10153157898526444983342442522509349751544628069655020714501469106560485335568031744363086428812458613298611705486575484949060071476336771787013086827249370837760079228966326381157362601117941598135887970204116315695922005257739717535113467576116457537034573566394606200326352750108986613704842427141311154365884025368007140877412082696197452204765601102060812248182357147029003119477666798838393581143284318605801960412230348678440113535106323752914052999383
1150542106513726817650139972729909828255218317628993460597028107105393555246147765465939841447131309663116387968301167984734271839502560981287912163869805061386856000512693327199196533234268250420494430092096092600298870795072161788783819981094475622742230718200687792119902466803146201147501691454598058376700402293886445799119016360465840775986651484396000021622048554205830892170459488294634754593666149042822041590871219650747119120404169499601869360969718087781144176112458803924840105582621089800886297913924753138105960912153892258110595224535043418
121667889514952984961507131922677260843106553082374496123888712368450523574265056727052708915954104749827609578384007726852461536675765162935020689912887988613489937291487047578107126309182713724507850328528771014390492327824647382362988912254163241325408077723376053333268627701001325994626534637859327199674332483036116700684908188251218520126186602504257771309167843420241202469964677498375644407683422974721304600236482956432364174350588238569004147104292003251368300413558549021765869852973446150899661801937169409528248660192705923908885774753990665388076810582448731153682292094412850133154229269332269150716981984428242953648112261572502821174335386037006377
13550403539934485037133955704051910807460554335434934298565855546232795111952584586401993132952971142182015480609264512861636605577507550166218366555413154258296891128713822913174257032158717616153022159217986085212375449169488763994662034167749214245405317830597187268553403329344881958397796072466369085226996247924758372511083264701369310544648860729716569783270413689603515263350070761086991344154319688605316773077523265694236733219615741493168274707290610272315084217983067802391982104629220340823058224005062246183425827517418733290968610357098357305908117108004446175728221154547256216004450613590407982300205857943504301679352047056790752636585778299775115584408486990188331316769695448354431284163558051301048391958283417280033254186582832913785286204518292739761827078197727
14181633168178380062252930854739286497268710590521561350657082848988699929256430111709011419559956534531164467369450213734585334727139729927622438017461960844799514575858265828555602449410805381764081852479358618164131217099806596021703996426083700337948643445009528799288164517688899506060386436140487636015877093878697407572148536703016343505120096981411505956473392142049218993414939023202446573483970895702505576421393590906670451476165275238062563131129715616492545764600633937465921500763652733668215786801111290208206070885430199932802369615000061673594703405682688504911381572106245231817773214924504409821286300021036677844214666791305882368082877501515057933383414041256310403285527129987665258160987911229281924819796050048538131484373897019933708111512324296754734791765619094483484585476174844235228872156312420477196430053295025629199285279254530742287679245496117149730820961576562499541378317674148
Technical details
Node id
n-2a75e568b1aee671 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-63c8a826cb5a9f43

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:              1;
  1:             33,                 45,                    9;
  2:           1089,               4995,                 4401,                  1134,                     81;
  3:          35937,             438615,               945027,                653184,                 177876,                  19683,                    729;
  4:        1185921,           35829135,            161204985,             225501570,              130268655,               35311302,                4675806,                 288684,                   6561;
  5:       39135393,         2848038975,          24997811679,           63441414540,            65591738550,            32737297041,             8618680503,             1242784620,               97332435,               3838185,                 59049;
  6:     1291467969,       223908132735,        3704470120701,        16134449124654,         27560628419121,         22584142245168,          9948364675704,          2515555424448,           377391488742,           33749160705,            1746846567,             47829690,              531441;
  7:    42618442977,     17522950411935,      536448122592867,      3883924958644584,      10477612416964896,      13261657870542627,       8997317818048161,       3551650313738472,        858215425063518,       130618578717900,        12641272275744,         769388393340,         28280101374,          569173311,          4782969;
  8:  1406408618241,   1368707962064895,    76743514727532945,    905841345302827290,    3752650189854881235,    7069442124710364714,    7040085434017534962,    4074442674770382948,    1458127300293687807,    336019344648290550,     51107149672713954,     5187785051593620,     350534088582930,     15456470433516,     424325877804,      6543101592,      43046721;
  9: 46411484401953, 106822509428882655, 10907076835869797799, 207325321721816410980, 1294969695760944574650, 3543628294943886294837, 5022117492236640700731, 4098803432410374151860, 2068030654655331261645, 677173052116483979355, 148760694571658476635, 22398340486514369760, 2337886644586214976, 169432445801056662, 8443535893024338, 282131997545448, 6005404999989, 73222472421, 387420489;
%F Formula
The row polynomials satisfy P(n,x) = (33+45*x+9*x^2)*P(n-1,x) + (45*x+18*x^2)*d/dx P(n-1,x) + 9*x^2*d^2/dx^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[(33 + 45*x + 9*x^2)*P[n - 1, x] + (45*x + 18*x^2)*D[P[n - 1, x], x] + 9*x^2*D[P[n - 1, x], {x, 2}]];
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] = (33 + 45*x + 9*x^2)*v[i - 1] + (45*x + 18*x^2)*deriv(v[i - 1], x) + 9*x^2*deriv(deriv(v[i - 1], x), 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

  • Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
    by Ray basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
    derived from Ray basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
    derived from Ray basis map theorem

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.