Real-rooted atlas

A027467 → x ↦ x+1 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(1 + 16\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
111617
2132256289
314876840964913
41641536163846553683521
518025604096032768010485761419857
619638408192098304062914561677721624137569
711125376143360229376022020096117440512268435456410338673
811287168229376458752058720256469762048214748364842949672966975757441
9114492163440648257536132120576140928614496636764163865470566468719476736118587876497
101160115204915201376256026424115235232153603221225472019327352832068719476736010995116277762015993900449
11117614080675840216268804844421127751073792885837004807086696038403779571220480120946279055361759218604441634271896307633
12119216896901120324403208304721921550214758421260088115221260088115201511828488192072567767433216211106232532992281474976710656582622237229761
131208199681171456468582401349517312287897026564606352424965527622909952491344258662403144603255439361372190511464448365917469723852845035996273704969904578032905937
141224232961490944656015362099249152503819796489212704849921289778678988813757639242547211006111394037766403555720167424256142228806696966305039478318694472057594037927936168377826559400929
Technical details
Node id
n-721a13439fc70d0d · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-d75da96e6240dd31

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: 1,  16;
  2: 1,  32,  256;
  3: 1,  48,  768,   4096;
  4: 1,  64, 1536,  16384,   65536;
  5: 1,  80, 2560,  40960,  327680,   1048576;
  6: 1,  96, 3840,  81920,  983040,   6291456,   16777216;
  7: 1, 112, 5376, 143360, 2293760,  22020096,  117440512,  268435456;
  8: 1, 128, 7168, 229376, 4587520,  58720256,  469762048, 2147483648,  4294967296;
  9: 1, 144, 9216, 344064, 8257536, 132120576, 1409286144, 9663676416, 38654705664, 68719476736;
%F Formula
The row polynomials satisfy P(n,x) = (1+16*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[(1 + 16*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] = (1 + 16*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.