Real-rooted atlas

A075504 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(9 + 10\,x - 8\,x^{2} - 9\,x^{3} - 9\,n x - 18\,n x^{2} - 9\,n x^{3}\right)\,P_{n-1}(x) + \left(9\,x + 27\,x^{2} + 27\,x^{3} + 9\,x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 81 + 189\,x + 109\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
081189109379
17292754337513518209
2656137179741966318919612200737
3590494986361469664201293113163043334515690035
45314416908733285469115598428457798441304768446493069186739723
547829691009737905624417251476165951213308388917525302567706262331412648206941869633
64304672115640308631135051687838127880251721550633318202325347955799807148210099652473376695763285450259681
73874204892548365883223454723382297895630583923290268964753435051709881321445666782018648807826596137536039478779936534312784343645107
834867844014311990042574950760335489251984503905307338747741132613495817871474316297054221902412954445150507265492154934812191381018811392465959810690618540762991483
93138105960974826393245461066223801981796543836399527472290822029200682511313945670212476538021667343387817257648334421539696979906822024162268944935346345955866850927435306413874932165686685892241
1028242953648113189459353662723444301484067761723338734858626571547151119632046190049934088905318342980888121322452432155261186116038381971989209854187232872484113131004934913418050101502228624521444465223931237109578131789458413089439809
112541865828329234642458908414852717544134112844461989710587759355225035404368903657070065661364549665111493600159831856462022571841839012544222245191104806580258031908224832045986125210408490210381638374378564744011631760882577584313634662181812390963020066105984904524367106323
122287679245496141963662959883461121424450489064312312635270092016236872715614404355782282792580131058865535696946395777988355231782031135635852182464159812147394484318046885920114057299773190570794039766042168234634067514817087693538668978941600804550172332208873217627424110859306129581910640764287936651882425476052840395
1320589113209464975269222535312682228669280186129460057352820414355026712615230573966736782412920795331125995988432688722714453093438552686963555750097389642394935626846962893032849503627859624109488027302313574281754374890454184524033512693679875617144314812307178051838868982805170561192835485985712481314707963790933892287112727516544416978299440632185527498742321409
1418530201888518411352272366484445167169350470160987807397010055129759820945729495753537298305680423919813544792732574999727086971148178280810914089584992269456985220580258438279747174425252704956978807766256921023932733703744623631873739861811425104626648365632380969020580916952763115313673043559798595114290580934827247479657332366450302207869436740056190526679969874913041451311822482205682277230703396752009332917637185
Technical details
Node id
n-b25768308db07696 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-1f4d1b1758c360f5

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:          81,           189,             109;
  1:         729,          2754,            3375,            1351;
  2:        6561,         37179,           74196,           63189,            19612;
  3:       59049,        498636,         1469664,         2012931,          1316304,           333451;
  4:      531441,       6908733,        28546911,        55984284,         57798441,         30476844,          6493069;
  5:     4782969,     100973790,       562441725,      1476165951,       2133083889,       1752530256,        770626233,        141264820;
  6:    43046721,    1564030863,     11350516878,     38127880251,      72155063331,      82023253479,      55799807148,      21009965247,       3376695763;
  7:   387420489,   25483658832,    234547233822,    978956305839,    2329026896475,    3435051709881,    3214456667820,    1864880782659,     613753603947,      87799365343;
  8:  3486784401,  431199004257,   4950760335489,  25198450390530,   73387477411326,  134958178714743,  162970542219024,  129544451505072,   65492154934812,   19138101881139,   2465959810690;
  9: 31381059609, 7482639324546, 106622380198179, 654383639952747, 2290822029200682, 5113139456702124, 7653802166734338, 7817257648334421, 5396969799068220, 2416226894493534, 634595586685092, 74353064138749;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (9-(9*n-10)*x-(18*n+8)*x^2-(9*n+9)*x^3)*P(n-1,x) + (9*x+27*x^2+27*x^3+9*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 81 + 189*x + 109*x^2.
%t Mathematica
P[0, x_] := 81 + 189*x + 109*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((9 - (9*n - 10)*x - (18*n + 8)*x^2 - (9*n + 9)*x^3)*P[n - 1, x] + (9*x + 27*x^2 + 27*x^3 + 9*x^4)*D[P[n - 1, x], 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] = 81 + 189*x + 109*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((9 - (9*n - 10)*x - (18*n + 8)*x^2 - (9*n + 9)*x^3)*v[i - 1] + (9*x + 27*x^2 + 27*x^3 + 9*x^4)*deriv(v[i - 1], x))/(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

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.