Real-rooted atlas

Veronese parent of A098172 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(2 + 10\,x + 7\,x^{2} + x^{3}\right)\,P_{n-1}(x) + \left(10\,x + 14\,x^{2} + 3\,x^{3}\right)\,P_{n-1}'(x) + \left(7\,x^{2} + 3\,x^{3}\right)\,P_{n-1}''(x) + \left(x^{3}\right)\,P_{n-1}'''(x)\)
\(P_{1}(x) = 1 + 3\,x + x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27x28x29x30x31x32x33x34x35x36x37x38x39x40x41x42x43x44Pn(1)
11315
224611575161255
345725258115229191319951938130304
4869042030441158836225411619332408374591969532587618736916618379
51682928747533610170986442188239673584073264292456831232278089916277159164831758593120410492010912329955413
6329952962711024808466533200704993121762300987020163631616085243148064202921627246698225297181403811255806419159509327915210768497195334075101068615811216014303146
7641194387297835766086892611181121117107947793665311538383664175921122288192253282904813136214780823819264114636891066880403487806465389677219661382161729951171119094477521516017651471952159194395925391120070204402161887654671626505
812814332710435249541126455571486258496172403739839801617608995740899200783176350818831201797084645973122082377040066154661281956548615382105601058501035975252563917467819774624810214618932419720191714512595603226290819560245526589829890081199110540007211019636268544690801191304901992657844356942831868222074880332913
9256171992652812693274656896446123013204646426250019638147065646142082376187432323319858463787989184011831164240681650630423656757594708659755228926466425432341806423011339435473355100812468201039605620894447632188118438349064131724351690270285362690188579591613328411829197501075864477587324082197214223462710244215285616315250061147521266478225784948915301701851538336798413156530565820335911100977978580028549018
105122063912089645699916588672035745693184788608039687480580754317056118962688828622070169613664826296276921642688743578206811202015514242201392396645198225314883901416522764396152288324433977169173614602525603401239152509993973782561830941304792563659091207128522918712005816734420551352197576552033256446886668527125704450474348741943248955465095666774011352559630959447359975844569975366468543916097146465884273586240224645022884212584800168966520247823993396455285309296920691261110469458996544411763303973050158447840143
111024247669455872164524653108270082861666271752111155259781500133522952322563038398186352732892311045523914092088590362081280453706442298630672856291841960553244519383926753073924950918808121619678509292807880420208155968195472935368364239658722261519384822086180177840261330776904099363287229590321834623992090401292462907006338488416390563836533933996485682829686487444597252367502589145881551144078784163310565844216422221692393283943354239444225174948560944102113013093260739031991149282711926563606543845695003487746047138390354326056667013919992717291485400632050054148905996273344815850053711820952564162017724513322153813491708211573773704001504083
12204829720334807045922946952567203842290056928352059872256898725882275367472854016771778634805067486485401602208025512174582093558740736271565817235054167478036820481695599452479789058441774796806027330149369685529862210501121324632441972367547906417479168191379249212742346593830888140819050761627976998251642209094401355432708241324408945056462720709654988162518889506004677456279898441313910283593714548016847610091330149452065655439602001530146324997549076092860037322963646734153935816731005552221870386990885179352606642899075061414635440894064322891353414269577694005063638988619140900108753250165681050354418058517030615011606531916767281378092028638965725338801553586942417143084242617962116188925413452002916528976605241081118590644150893823487660019045564118399360111987784791583144740504
134096356644017889282132268035804548505618323061526447642028646413496919402612377549000601619540528278582936469877906432876367979741527285819866089472160501665787016965757874698926081437534735898310263118574371985927129412928663496172306866860684802140811841886101461709185294943488415962765334403600806458526042982455024378256582195463375949269299205156362211802442768128167575150848353317561231677062807335391222412818160335484382503817449980486071047149689717549722360139506111515842193405724571489604845669665737285318521445346976533378412727821610313624575441216031647729844760161507873549517297568007023210020586421576314915123942327474021494860834954699942992744840184755717195646667897722194013119966824555332662738052577150160711408783214902513760428439211954148294011518658065107936242269050718677395480677091613955948958727035689314562735690354437575849048490487284583714851922979792177806872890348257581914264394753124181742478057901824056463340302417
14819242797282150809676761734883528042496014659387419450511642152960202582055241819599919140495364937982915704731527489983102976346274486615597418763381144374272940657768419594109897055082255257612021798132720114507642075861587302482637421052621896561560188048600217633638865373554378511791821297770984968710347698083243739026916636164422400151535469357019533383191252583675013121848663013260264729967888475471340364816363395718286633964529245850724836864108030211628340477605456269529145297925441179074457202599771541929800711488213030788323155830437911638170627424065851494444341992302801293304511033616282972166307014970382503555670668832561471743117436129098784380256624531450022891518129817899916365153671339428646776794841690632557746079257639977851542442571332607260732496443382454716088068776831056538541506170723857711748458236552552207138917464456958787198303665303490387709538887190350312255114193294388876345602160362399768209494111081637675920606400769829699076030397310487866622328917498103609812205146524963725451842200128542287822482658985215021793553519921635579163491454636358008874182268686363817064134273860583967518579
1516384513567385817907227634234829417811550208117278476872366941192602419230397569957256814974335251701761246438266272025233403672529874944136431301282077197445784364238225408548201190751648302633506417589746483299586365853283083369260535389926912000944138576613373296197514027337381416960517898313834415226724733232843533439846417746820177640334822929651322101320048128402814939888166214250015925202449714163206340060632540538946466335492473670647270471769713339282896727380098303466226096896601855602138407483243441092028042950456323830787455406395623203695596024088189376018883180662391301319452793294215983367040733119335689785007086961363981312607891222738163327149123509389458517383309930885702154159513228738479845076495398785601168035382718216413800126955622102094401971358054747006379380644099051633496027615103675893189222770642941643999843230495891497364864156813024892746163172096831700750910864501284447650426252730224353979867632148757611141837210211416769594612430550960781089758414353409658934929016723154871811413647489007531833225523469313899885041736584277215852457246065675001595291762262618748424903239451801196845999349300672800175301504598997983672275048027734027592763574872442569314374270445715388182399197767055501015673768645681481569642927426475414051198047451010041327118438360713179234442373799131477010832
Technical details
Node id
n-3537990b4764b63a · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-f25876d70728b9b9

Observed on the stored rows ?

  • Degrees: 2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 32, 35, 38, 41, 44 (affine: slope 3, intercept -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:   1,           3,               1;
   2:   2,          46,             115,                 75,                   16,                      1;
   3:   4,         572,            5258,              11522,                 9191,                   3199,                     519,                      38,                        1;
   4:   8,        6904,          203044,            1158836,              2254116,                1933240,                  837459,                  196953,                    25876,                     1873,                       69,                        1;
   5:  16,       82928,         7475336,          101709864,            421882396,              735840732,               642924568,               312322780,                 89916277,                 15916483,                  1758593,                   120410,                     4920,                     109,                       1;
   6:  32,      995296,       271102480,         8466533200,          70499312176,           230098702016,            363161608524,            314806420292,             162724669822,              52971814038,              11255806419,               1595093279,                152107684,                 9719533,                  407510,                  10686,                   158,                    1;
   7:  64,    11943872,      9783576608,       689261118112,       11171079477936,         65311538383664,         175921122288192,         253282904813136,          214780823819264,          114636891066880,           40348780646538,            9677219661382,            1617299511711,            190944775215,             16017651471,              952159194,              39592539,              1120070,               20440,                216,                 1;
   8: 128,   143327104,    352495411264,     55571486258496,     1724037398398016,      17608995740899200,       78317635081883120,      179708464597312208,       237704006615466128,       195654861538210560,       105850103597525256,        39174678197746248,        10214618932419720,        1917145125956032,         262908195602455,         26589829890081,         1991105400072,         110196362685,          4469080119,          130490199,           2657844,            35694,            283,             1;
   9: 256,  1719926528,  12693274656896,   4461230132046464,   262500196381470656,    4614208237618743232,    33198584637879891840,   118311642406816506304,    236567575947086597552,    289264664254323418064,    230113394354733551008,    124682010396056208944,     47632188118438349064,    13172435169027028536,     2690188579591613328,     411829197501075864,     47758732408219721,     4223462710244215,     285616315250061,     14752126647822,      578494891530,      17018515383,      367984131,       5653056,       58203,       359,        1;
  10: 512, 20639120896, 456999165886720, 357456931847886080, 39687480580754317056, 1189626888286220701696, 13664826296276921642688, 74357820681120201551424, 220139239664519822531488, 390141652276439615228832, 443397716917361460252560, 340123915250999397378256, 183094130479256365909120, 71285229187120058167344, 20551352197576552033256, 4468866685271257044504, 743487419432489554650, 95666774011352559630, 9594473599758445699, 753664685439160971, 46465884273586240, 2246450228842125, 84800168966520, 2478239933964, 55285309296, 920691261, 11046945, 89965, 444, 1;
%F Formula
The row polynomials satisfy P(n,x) = (2+10*x+7*x^2+x^3)*P(n-1,x) + (10*x+14*x^2+3*x^3)*d/dx P(n-1,x) + (7*x^2+3*x^3)*d^2/dx^2 P(n-1,x) + x^3*d^3/dx^3 P(n-1,x) for n >= 2, with P(1,x) = 1 + 3*x + x^2.
%t Mathematica
P[1, x_] := 1 + 3*x + x^2;
P[n_, x_] := P[n, x] = Expand[(2 + 10*x + 7*x^2 + x^3)*P[n - 1, x] + (10*x + 14*x^2 + 3*x^3)*D[P[n - 1, x], x] + (7*x^2 + 3*x^3)*D[P[n - 1, x], {x, 2}] + x^3*D[P[n - 1, x], {x, 3}]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1 + 3*x + x^2; for(n = 2, N, my(i = n); v[i] = (2 + 10*x + 7*x^2 + x^3)*v[i - 1] + (10*x + 14*x^2 + 3*x^3)*deriv(v[i - 1], x) + (7*x^2 + 3*x^3)*deriv(deriv(v[i - 1], x), x) + x^3*deriv(deriv(deriv(v[i - 1], x), x), x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

  • Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
    by Ray basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
    derived from Ray basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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.