Real-rooted atlas

A200536 → basis A007318 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(15 + 13\,x + 2\,x^{2}\right)\,P_{n-1}(x) + \left(13\,x + 4\,x^{2}\right)\,P_{n-1}'(x) + \left(2\,x^{2}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27x28Pn(1)
011
11513230
22255593416841197
333751857725298127672746252863023
4506255640311460969141103461965713540815032800164257572
575937516450993754333821249366559458657537041081800767497900066560232032358263647
61139062547049967936756878219862822238118816614647437813705263670294555637610071529708559352025720063366436610341405
7170859375133220691371734272277387290767113071335153124816125683147153267262672577821814509537544608188538587540058050073189627213648903392165761284449765739816
8256289062537523910771180310421422895178767886703414007270659640719099737062611814705763385422268848419918100205936939142094065269235040539848871696485763505922945617248117755904295603241984256632423108745467
9384433593751054001259403336778108701534235873891684277351405737035623346527085368324164031290828766235506741890432697829745078740437785328232186427374846645415836463646578551043891937445129893230016335242281696152037022084659279369156864103680512103632583814264807
10576650390625295620116304799167294060165077012445121679687407581369640993801969374366508271756730967553716120188131330754722059297596218500265214727219837891199559457944824595525802987960968910147241646438197156620412666639352458941243713968040579811708960201307461542470857419136171780480027152640250880102419349782380817571698
11864975585937582848597116124977578499660208093781648971185251616134713078359344723074738648001853926427175436294554365054573451849811032710766745020581258169936751168967530914035748497718353715751379481398105097674423352411513184853923159076287060777990347652835883476805567348533611559280852470688899147202779268359964801082205824832030423545011259809164807769344059699220484075665428849453843297
1212974633789062523208851875132179134244108385545553209110425240127192687083412315065499942943664557614279283453873345203281607703575161128107189122455666567168761117369002358507403967492033841421503579160706359205600456624230042746940113200720790516629339436246894475658932928022885084180699424115489237531210278396421560239949699067087693175384647253892669004856164033570958524800016943871126533125316207674624012211851773441986072985621580390414008324096960137252047622703013557
13194619506835937565001652274295882731544930641660998146102736002489370811327360151148345758743377136351955773069680574736936245318126649616466362927966493474527290238609934098956514006487129467207321218438752652603201466782980450332610293679870071822138821127641167037304068745700244239333866604069641862184528077320021259575994505498411196296082585814082142181494730853785639954816814387063232224264949121071302445664659093606176838314200560864274985648347970560938412571427302424229727826304046332209331206337515724858455654432481288192251100482508004207603195094
1429192926025390625182029926903917143519696323855760013562939814890072973652690580821678106342560130357669722720484961132844191116422714060445433480100546067792210716040511020011736455670953658930900172004582804823676762482128541754314991279738397238054183012251895238550731978121721242607311072369408204357586626621971445143629704819304129745930000853811596008032271491214656692342365067308215771913525160060684778322656423418232850060458093678362242365947256830485053601687322024485527455188899942816917715964192046539969445643081408169600631800404147247946783063132928103658532928204816757333791744195469094912154969292874547201638472429191690572832445348208041
Technical details
Node id
n-29feaac19fa9e41e · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-669843b93c09642d

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:          15,             13,               2;
  2:         225,            559,             341,               68,                 4;
  3:        3375,          18577,           25298,            12767,              2746,               252,                 8;
  4:       50625,         564031,         1460969,          1411034,            619657,            135408,             15032,               800,               16;
  5:      759375,       16450993,        75433382,        124936655,          94586575,          37041081,           8007674,            979000,            66560,             2320,              32;
  6:    11390625,      470499679,      3675687821,       9862822238,       11881661464,        7437813705,        2636702945,         556376100,         71529708,          5593520,          257200,           6336,            64;
  7:   170859375,    13322069137,    173427227738,     729076711307,     1335153124816,     1256831471532,      672626725778,      218145095375,      44608188538,       5875400580,       500731896,       27213648,        903392,        16576,         128;
  8:  2562890625,   375239107711,   8031042142289,   51787678867034,   140072706596407,   190997370626118,   147057633854222,    68848419918100,   20593693914209,    4065269235040,    539848871696,    48576350592,    2945617248,    117755904,     2956032,     41984,     256;
  9: 38443359375, 10540012594033, 367781087015342, 3587389168427735, 14057370356233465, 27085368324164031, 29082876623550674, 18904326978297450, 7874043778532823, 2186427374846645, 415836463646578, 55104389193744, 5129893230016, 335242281696, 15203702208, 465927936, 9156864, 103680, 512;
%F Formula
The row polynomials satisfy P(n,x) = (15+13*x+2*x^2)*P(n-1,x) + (13*x+4*x^2)*d/dx P(n-1,x) + 2*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[(15 + 13*x + 2*x^2)*P[n - 1, x] + (13*x + 4*x^2)*D[P[n - 1, x], x] + 2*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] = (15 + 13*x + 2*x^2)*v[i - 1] + (13*x + 4*x^2)*deriv(v[i - 1], x) + 2*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.