Real-rooted atlas

A038337 → basis A007318 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(34 + 11\,x + 11\,n\right)\,P_{n}(x) = \left(1035 + 748\,x + 121\,x^{2} + 253\,n + 121\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1035 + 748\,x + 121\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
010357481211904
1296243415512342133177452
28151891303456751410181016146413065712
3218275984483539535845040137758502488970161051118933904
4572834527144062146814795680357885908802273015253285440417715614543542400
51480358890044108258579554639265183797557956515180374440350044348542163151819487171171473290176
63779356246171302715823200194076337747616269418445288354627504302671745901445133983778053005105122143588816407848717568
7955394020428237415626837083644844332484006404519145670840266810652068165421624585144408380737376726091991460655938175862357947691237510589351168
82395533039745791050933422471020205785947603956523644292191080001761242765059470885869834345496303272978406090692745544444809983764882575801702214940259374246018743246958370816
95965417614765456289859497809240596358147205949671083000332200262455715239838780170004262207491443917417865041659300836524229005530527010477776359727601342261723101759700596800774285311670611319978551879265280
10147685512476658685787435125149040192191307268554098789427975847706178552402739010962608661015188823317437964880093768564811766182833688364271824433686497785912536955153674053276004801771785474494310116407161609288313842837672111651348043913375744
1136378276557153732862111902828416219195556301611448156373728911897980107870896141000136555495883248307833571353060770502945002890656048296400191555782401179487726021795113588617506137435370949919828521975389618468686423033211168426030138718534251068234522712143931422396461013678886912
1289214436062754726791560225458980167486044162616517788048878053528901493876720271847043510807223415302950139308042059092732040476420106166472855977624119900635944330625208003687448811222705138611030396052769607884362116504712628765358930765545465856160962955928766614673851643281098051115637974983358324115254001010166301949952
132179453229013284964092147203819503545299205154621860036586381759863089439084783426882929425119218662241471121369040115856638372704997354587847115662505013111310104739310668601766154254520420061635181914491500676032282057495942078517001534870698530091943174757068943148038354230262501555222132037517557422417014251936724151274529104177248169415651548981759759495739326464
145306041185741385787579338358376830633815368019212953936116311986330053202711491221464010632452960393531973047078284889470041964969108997834721902083398461392266013255734552212989080858403296557968010426483470923885539359903993256497546685500560001189816816421192858058162720245579311764814710446563103677832837895805727124617490146886404716492933103853220022724230041621141444594972986357216119697134495893465220055040
Technical details
Node id
n-85e6c376a9649519 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-024b608e52614862

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:             1035,               748,               121;
  1:            29624,             34155,             12342,              1331;
  2:           815189,           1303456,            751410,            181016,             14641;
  3:         21827598,          44835395,          35845040,          13775850,           2488970,            161051;
  4:        572834527,        1440621468,        1479568035,         788590880,         227301525,          32854404,           1771561;
  5:      14803588900,       44108258579,       55463926518,       37975579565,       15180374440,        3500443485,         421631518,         19487171;
  6:     377935624617,     1302715823200,     1940763377476,     1626941844528,      835462750430,      267174590144,       51339837780,       5300510512,        214358881;
  7:    9553940204282,    37415626837083,    64484433248400,    64045191456708,    40266810652068,    16542162458514,     4408380737376,     726091991460,      65593817586,      2357947691;
  8:  239553303974579,  1050933422471020,  2057859476039565,  2364429219108000,  1761242765059470,   885869834345496,   303272978406090,   69274554444480,    9983764882575,    801702214940,   25937424601;
  9: 5965417614765456, 28985949780924059, 63581472059496710, 83000332200262455, 71523983878017000, 42622074914439174, 17865041659300836, 5242290055305270, 1047777635972760, 134226172310175, 9700596800774, 285311670611;
%F Formula
The row polynomials satisfy (11*n+34+11*x)*P(n,x) = (253*n+1035+(121*n+748)*x+121*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1035 + 748*x + 121*x^2.
%t Mathematica
P[0, x_] := 1035 + 748*x + 121*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((253*n + 1035 + (121*n + 748)*x + 121*x^2)*P[n - 1, x])/(11*n + 34 + 11*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1035 + 748*x + 121*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((253*n + 1035 + (121*n + 748)*x + 121*x^2)*v[i - 1])/(11*n + 34 + 11*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

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.