Real-rooted atlas

A038337 → basis A013609 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(23 + 44\,x + 22\,n x\right)\,P_{n}(x) = \left(529 + 2024\,x + 1452\,x^{2} + 506\,n x + 484\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 529 + 2024\,x + 1452\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0529202414524005
1121676982810018842592224775
2279841214139246086483918464117128012119625
364363436156502017666484022531168013469720030921792635596875
414803588916991945526094936980103643372809294106800426720729679365932832661478125
5340482544745595053812196256970756417164575520498783731600343510187328127779151808199548631041652449359375
6783109852811198498557344601854710318415351656379136229440516536002106862482278411755681966336367169481113649388286182482580949140625
71801152661463310111501712761779770357655845296321450801929498837384590401090301334579072811142055677184380020412952576102233752397568120726921779204086075469921875
84142651121364979250717104372051168397782605401740219905263488036412209974263200501538613906373124664066820143808029134898326364160117568815257203202776719200921600292159150705664200520370283203125
99528097579139272005043142740611614384005154443518055034454503957808013160413033555128002114821155305206656236001781099276684818427823191425331209914970086690803203512549789165824007391626512853299270118196169359369772530498896484375
102191462443202032150308355217855345639699854226264109681687723271454706112045403424965765191600833843769806052910081085608193056672750081017215840166678282246841329359816654208032315458060325580800102004445877375528961935262214274318336167115034203639808473473995077197265625
1150403636193646738312535165175115623612107911421942299716312504629258164957127488150840267386264358760031164910896501227514244637097853199216461056506912560349728010649640911149571703592163842415580490009337164800101664431057784277196828932170103401059123249967395226888302592395934081036315852822823285296682373046875
121159283632453874980931048639895286390792828956231554517090543721477187464810147227737648570566098377123520720111501125651926613995392186643188591268462557504233179777760874884898816219556502701475944612608155563383556601313413120818398670015163431434243105386257765047012556880447506315290167173121274907740936937046016933273191014173081601095310836670089111328125
13266635235464391245607765127197419557487394076845383740833817212180424691396132917327974560152334957308546432860440038467888349914681828410247154655562665291064704320100558779159377294112614401082099906171560012733568089448945545045755212544005646950823104627676896256267839564732235304833024092514632262583692249088021992158531162164043776032197925089988971315200219008108824659283148852360822574526214599609375
1461326104156809986489611877112057669314369073282019935801187631766720160120220333674548905149721604671605357462090607720160012869257193426002648049971226329132470608271118111897641117367478500938037157888049776595683891760585744128047024588515109768454594560034634631715041716418297036819712991964292518435710566485113461681576996869160960269770477982255878936985605924418216557970721996800805949840474746161987584511931454377641074360322494469587450428863525390625
Technical details
Node id
n-ef17a68c74d5bfac · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-bd76ccacb5cfb628

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:             529,              2024,               1452;
  1:           12167,             69828,             100188,              42592;
  2:          279841,           2141392,            4608648,            3918464,             1171280;
  3:         6436343,          61565020,          176664840,          225311680,           134697200,            30921792;
  4:       148035889,        1699194552,         6094936980,        10364337280,          9294106800,          4267207296,           793659328;
  5:      3404825447,       45595053812,       196256970756,       417164575520,        498783731600,        343510187328,        127779151808,         19954863104;
  6:     78310985281,     1198498557344,      6018547103184,     15351656379136,      22944051653600,      21068624822784,      11755681966336,       3671694811136,       493882861824;
  7:   1801152661463,    31011150171276,    177977035765584,    529632145080192,     949883738459040,    1090301334579072,     811142055677184,     380020412952576,    102233752397568,     12072692177920;
  8:  41426511213649,   792507171043720,   5116839778260540,  17402199052634880,   36412209974263200,   50153861390637312,   46640668201438080,   29134898326364160,  11756881525720320,   2776719200921600,   292159150705664;
  9: 952809757913927, 20050431427406116, 143840051544435180, 550344545039578080, 1316041303355512800, 2114821155305206656, 2360017810992766848, 1842782319142533120, 991497008669080320, 351254978916582400, 73916265128532992, 7011819616935936;
%F Formula
The row polynomials satisfy (23+(22*n+44)*x)*P(n,x) = (529+(506*n+2024)*x+(484*n+1452)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 529 + 2024*x + 1452*x^2.
%t Mathematica
P[0, x_] := 529 + 2024*x + 1452*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((529 + (506*n + 2024)*x + (484*n + 1452)*x^2)*P[n - 1, x])/(23 + (22*n + 44)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 529 + 2024*x + 1452*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((529 + (506*n + 2024)*x + (484*n + 1452)*x^2)*v[i - 1])/(23 + (22*n + 44)*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.