Real-rooted atlas

A038322 → basis A007318 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(8 + 19\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
181927
264304361729
351236488664685919683
4409638912138624219488130321531441
532768389120184832043897605212840247609914348907
62621443735552221798407023616012510816011885275247045881387420489
720971523486515224841420898330624023353523203327877056263456933689387173910460353203
8167772163187671042649751552125863198723736563712070994710528843062187525720779129616983563041282429536481
913421772828689039362725458739215103583846453806517452812779047895042023349250048205948048665612228165389523226876977797625597484987
101073741824255013683202725458739201726123868160717420232704020446476632064404669850009605491947964416048912661558080258150158223206131066257801205891132094649
118589934592224412041216266489298944018987362549760901899721113602998816572702727122189360168961208228552171520143477140570368011358606961820805395338306864881164902588982195559060566555523
1268719476736195850508697625582972698624202531867197440108227966533632041126627282780161139550297627033623197988201693184344345137368883203634754227782656025897623872951424111830648542290242213314919066161150094635296999121
135497558138881697371075379224187537824153621063314188533761250634279944192053464615467614208169304615647444992402098462162681856716237885727277056945036099223490560897784294262316032581519372419909248230184751582880744420529834622570594052555153018976267
1443980465111041462350464942082257503530254336214462835374161921400710393537495046653374369303101442370264619064229888643357539460290969613369773866909171712211688086226061885442513796023934484889621710056570343278592128903460886413216644709934147772790608799006685782884121109418989131512359209
Technical details
Node id
n-7d8dce85fe57eb21 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-fa8b89dcb83e73ac

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:         8,         19;
  2:        64,        304,         361;
  3:       512,       3648,        8664,         6859;
  4:      4096,      38912,      138624,       219488,       130321;
  5:     32768,     389120,     1848320,      4389760,      5212840,       2476099;
  6:    262144,    3735552,    22179840,     70236160,    125108160,     118852752,      47045881;
  7:   2097152,   34865152,   248414208,    983306240,   2335352320,    3327877056,    2634569336,     893871739;
  8:  16777216,  318767104,  2649751552,  12586319872,  37365637120,   70994710528,   84306218752,   57207791296,   16983563041;
  9: 134217728, 2868903936, 27254587392, 151035838464, 538065174528, 1277904789504, 2023349250048, 2059480486656, 1222816538952, 322687697779;
%F Formula
The row polynomials satisfy P(n,x) = (8+19*x)*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[(8 + 19*x)*P[n - 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] = (8 + 19*x)*v[i - 1]); 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

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.