Real-rooted atlas

A130123 → basis A013611 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(10 + 8\,x\right)\,P_{n-1}(x) + \left(8\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
110818
210022464388
31000483234565129800
41000094976128512450564096282640
5100000178956841011202560000532480327689115936
6100000033012224120945664119848960428441605898240262144323811392
71000000060222003234086850565042429952275824640063766528062390272209715212523734144
8100000000109199605769344357171219871209881615618578841653949235200871995801663753420816777216522684924160
910000000001973592903682516892549120750375993344081494999040003946948067328937351446528111836921856634178764813421772823371124118016
1010000000000356046722662467018080600064275262978129920402309075435520262543402598400859419684372481488004841472013639876608006174015488010737418241112952822400000
11100000000000641684100792321770953833414656989508590121779219099084993331200163456427334041607084981390147584166961894286950421997547421696015974593986560590558003200858993459256166185420605440
1210000000000001155831381426176465581469494149123506005513087221768813222569296527369700748166168576005416940624957931521668747013505679362963513663501107230697205006336001809469721804805566277615616687194767362991172806137352192
13100000000000002081296486567116812197584717361971201229288392009187328039820339201515192320555543188862801018883917885415769086361615347282789103828992352799772179536281648879817413204377640842991501312000199307098377420851814485458944549755813888167493944713937494016
141000000000000003747133675820810243188022398406649446442771612105701326848017707973178243730636803096278657926126632960271680809223631090483213263514973423796224003838500937256874803206830543205319049216075862546281744302085220648884203683842143690332884172847718804645478443980465111049830492487522746646528
Technical details
Node id
n-7f838cb81c71df94 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-c7f57c7d3a795bc5

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:          1;
  1:         10,            8;
  2:        100,          224,            64;
  3:       1000,         4832,          3456,           512;
  4:      10000,        94976,        128512,         45056,          4096;
  5:     100000,      1789568,       4101120,       2560000,        532480,         32768;
  6:    1000000,     33012224,     120945664,     119848960,      42844160,       5898240,       262144;
  7:   10000000,    602220032,    3408685056,    5042429952,    2758246400,     637665280,     62390272,      2097152;
  8:  100000000,  10919960576,   93443571712,  198712098816,  156185788416,   53949235200,   8719958016,    637534208,   16777216;
  9: 1000000000, 197359290368, 2516892549120, 7503759933440, 8149499904000, 3946948067328, 937351446528, 111836921856, 6341787648, 134217728;
%F Formula
The row polynomials satisfy P(n,x) = (10+8*x)*P(n-1,x) + 8*x*d/dx 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[(10 + 8*x)*P[n - 1, x] + 8*x*D[P[n - 1, x], 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] = (10 + 8*x)*v[i - 1] + 8*x*deriv(v[i - 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

  • Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
    by Ray basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
    derived from Ray basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
    derived from Ray basis map theorem

Proof chains ? JSON

Same fingerprint ?

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  8. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.