Real-rooted atlas

A038259 → basis A013610 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-4 + 15\,x + 15\,n\right)\,P_{n}(x) = \left(121 + 330\,x + 225\,x^{2} + 165\,n + 225\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1261541
24517802251456
367762029517550337547996
494501406560608850351000506251511536
5125912670875751524600015221250658125075937546154576
616266151113321340318940875457380000342478125118462500113906251378239616
72055010761707945855594937035011162930625120062250007192040625207309375017085937540467966656
82552819401246601291201024767513002379748140003348879187502881494000001438408125003553875000025628906251172644285696
9312963966263446306191351664558715600461145380850080316499725009041973806250648336150000027740728125005997164062503844335937533621157396736
1037962957825146944594939002584729643512583227935780000172929517818750240949499175000226049345156250138929175000000520138652343759995273437500576650390625955592647499776
11456498672977662638880411415387292908246750142160130393187534331523509250005706674088018750662611122731250053283059929687502865414234375000953587529296875164922011718750864975585937526962915269794816
12544945290867018216976113596805637499237027350232375744948050006397205867693437512359348463330000017120022264056250017038571727375000011988688484179687557308284687500000171645755273437502698723828125000129746337890625756090964319481856
136465162456045261062643317190669580115517107568800366437450406777750113283175662174375024949102884004406254016788250582250000476914905927281250041531518585476562502597549171572265625111751155140625000030428111162109375043854262207031250194619506835937521089799422231760896
14762951938380875113576841157695046011157754830502029755608086197529816000192379661463558318754757893377811323750087321860094015421875120503647517467500000125190162805911328125969068766994453125005454853260301757812521334311435937500000532491945336914062570841500488281250029192926025390625585552078076081831936
Technical details
Node id
n-6685c0882ca74bac · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-c33bca10ccffad3d

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:          26,           15;
  2:         451,          780,           225;
  3:        6776,        20295,         17550,          3375;
  4:       94501,       406560,        608850,        351000,         50625;
  5:     1259126,      7087575,      15246000,      15221250,       6581250,        759375;
  6:    16266151,    113321340,     318940875,     457380000,     342478125,     118462500,      11390625;
  7:   205501076,   1707945855,    5949370350,   11162930625,   12006225000,    7192040625,    2073093750,     170859375;
  8:  2552819401,  24660129120,  102476751300,  237974814000,  334887918750,  288149400000,  143840812500,   35538750000,   2562890625;
  9: 31296396626, 344630619135, 1664558715600, 4611453808500, 8031649972500, 9041973806250, 6483361500000, 2774072812500, 599716406250, 38443359375;
%F Formula
The row polynomials satisfy (15*n-4+15*x)*P(n,x) = (165*n+121+(225*n+330)*x+225*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[((165*n + 121 + (225*n + 330)*x + 225*x^2)*P[n - 1, x])/(15*n - 4 + 15*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] = ((165*n + 121 + (225*n + 330)*x + 225*x^2)*v[i - 1])/(15*n - 4 + 15*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.