Real-rooted atlas

A038281 → basis A013611 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(11 + 34\,x + 23\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 121 + 506\,x + 529\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
01215065291156
113318349174571216739304
2146411224523840545353482798411336336
3161051168371570409901472207015391255643634345435424
41771561222250381161763353238855405079114154247986381480358891544804416
51948717128522132117891155596234796645130363929851635474756311398763453340482544752523350144
6214358881358563946426240361532109732420952286800645670479739261848501545591932299624639336783109852811785793904896
7235794769144372288367371113684524181058494570856786527842661187354673073816551004533756148314196471327752787542819180115266146360716992766464
825937424601542327968930510281316220528452049146840104108634378210261218028076236455152624678290543818720394840426403314855045198126792760930414265112136492064377754059776
92853116706116562168424053686044880696454303372433459551799592108537630526789689953742611014693517214618164505162919439101719826703248681511986670962036265501260785685152995280975791392770188843638032384
1031384283767217874602108863690557924251931463116129024073402969326979087089599337484391277176242323257378721596434293630107319224567542812072064895527413522329595660330832118552200914125770888044638364219146244320203212386420683693101056
1134522712143931938390084639579117725301527510829025606450442496247179306445494866517756575334940795214950317972165312604103506648508911081721623172442526105599718855033423283044845157696643176549102348992618495191643026313379129377890590350403636193646738381138303245565435904
123797498335832411111631331034578215108080362697221912635849030619494687265613192606209441303834733508986940269529362096418226762722771462671960437978441661426024836710392358073502943711778122660713207622971404400946162108275317100866824130192962097575453177621599738215976982115928363245387498092758702310349224820736
134177248169415651131013692586218145191756404603464739517374292417101805185108984197889093141615501327310289828451429174704971767667490649546966141760918403330459820193277282938878185159702133095207389837152003535851558056345229719041869663568304107555132716061291536664992056403781978402818101015191281799354889371848526663523546439124560793795878551873643905024
14459497298635721611537227326344959568241065194358641386802352211896469167471201598434902373366077020802123696463725522286430748075031109478354312918448994435737665128802160442521002246553200704015367917822357230191205877038497903631946007126702738121576043045859365839506696827613259650203756885427329653285928801683279834323026472266804692780144173285922683261326104156809986489613189059870763703892770816
Technical details
Node id
n-e1c5805a0cc8733d · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-2f74b0bddfe64ff1

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:          121,           506,            529;
  1:         1331,          8349,          17457,           12167;
  2:        14641,        122452,         384054,          535348,           279841;
  3:       161051,       1683715,        7040990,        14722070,         15391255,          6436343;
  4:      1771561,      22225038,      116176335,       323885540,        507911415,        424798638,         148035889;
  5:     19487171,     285221321,     1789115559,      6234796645,      13036392985,      16354747563,       11398763453,        3404825447;
  6:    214358881,    3585639464,    26240361532,    109732420952,     286800645670,     479739261848,      501545591932,      299624639336,       78310985281;
  7:   2357947691,   44372288367,   371113684524,   1810584945708,    5678652784266,   11873546730738,    16551004533756,    14831419647132,     7752787542819,     1801152661463;
  8:  25937424601,  542327968930,  5102813162205,  28452049146840,  104108634378210,  261218028076236,   455152624678290,   543818720394840,   426403314855045,   198126792760930,   41426511213649;
  9: 285311670611, 6562168424053, 68604488069645, 430337243345955, 1799592108537630, 5267896899537426, 11014693517214618, 16450516291943910, 17198267032486815, 11986670962036265, 5012607856851529, 952809757913927;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (11+34*x+23*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 121 + 506*x + 529*x^2.
%t Mathematica
P[0, x_] := 121 + 506*x + 529*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((11 + 34*x + 23*x^2)*P[n - 1, x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 121 + 506*x + 529*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((11 + 34*x + 23*x^2)*v[i - 1])/(1 + 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

Candidate equalities ?

Same fingerprint ?

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.