Real-rooted atlas

A038326 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(23 + 70\,x + 47\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1234770
2529216222094900
31216774589152421103823343000
4279841228739670113669551716487968124010000
56436343657626352687690305492236705611633152293450071680700000
61480358891815048726927253153525264288820387202687353164961096610779215329117649000000
734048254474870380748129857551542710168876250052077987755445254779368276317354536679695066231204638235430000000
8783109852811280214368072915631580642837421464600184955874367504701562646792094641596617374531489321865416519223811286661761576480100000000
91801152661463331255467738632707653388472281291040528706348395731988146945880866971490897621101665988426721296481307060973724928936338984527111913047310276740353607000000000
1041426511213649846541750887610778450349185780542419903086065720151697262122995890371988068858129052633457943345364690739690020800798520566827678983220605257400008813636410525991322358300492824752490000000000
1195280975791392721417506297456533218831042604447185134152943509682839554827724738739943101568549690351777502632052971933275453314467853938156505063904780246759425160435532561101114925005865133075804556650023972472159215084012303197732674300000000000
12219146244320203215373847034634548286039736775882742306411402360096360707801891556503486528036956184567350529865581681474436708930670852444258255373862390795272832983702640033607004952995621302573100539580183644610288177033078668231594336318739562811619148310894857824113841287201000000000000
1350403636193646738313389835527964416131164171026908085449954123009305668811851632262841710504719098116452311482047260537261152962979510852895797840108128697261308091412977612197242541787400969889601223922692367339265333605183032461587216442968338102006233532796515646386347412534495756248940595460999706120583177327968890104070000000000000
141159283632453874980933165592615419553801444052558887002929070993600817856850674202324420235030782519549593496982699691024210333121248225349253118290558630643475920074020172204996970152105853497425905187174085871442062138845664868748416214734113157770923658951209109486690658534926793787645593341805381675607943499175844190537082778309929425666698618766740933436967822307284900000000000000
Technical details
Node id
n-3006e6e515fa7a82 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-17f3380a78a4ba33

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • 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:             1;
  1:            23,             47;
  2:           529,           2162,            2209;
  3:         12167,          74589,          152421,           103823;
  4:        279841,        2287396,         7011366,          9551716,          4879681;
  5:       6436343,       65762635,       268769030,        549223670,        561163315,        229345007;
  6:     148035889,     1815048726,      9272531535,      25264288820,      38720268735,      31649610966,       10779215329;
  7:    3404825447,    48703807481,    298575515427,    1016887625005,    2077987755445,    2547793682763,     1735453667969,     506623120463;
  8:   78310985281,  1280214368072,   9156315806428,   37421464600184,   95587436750470,  156264679209464,   159661737453148,   93218654165192,   23811286661761;
  9: 1801152661463, 33125546773863, 270765338847228, 1291040528706348, 3957319881469458, 8086697149089762, 11016659884267212, 9648130706097372, 4928936338984527, 1119130473102767;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (23+70*x+47*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((23 + 70*x + 47*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((23 + 70*x + 47*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

References

  1. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [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
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  6. [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
  7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.