Real-rooted atlas

A013622 → x ↦ x/(x+1) → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(3 + 8\,n x\right)\,P_{n}(x) = \left(9 + 48\,x + 64\,x^{2} + 24\,n x + 64\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
131619
2996192297
327432172820484235
481172810368245762048057233
5243648051840184320307200196608746691
67292332823328011059202764800353894418350089502009
7218781648979776580608019353600371589123853516816777216118694587
865612799363919104278691841161216002972712964624220164026531841509949441461537825
919683944784151165441254113286270566402006581248416179814454358179844076863488134217728017791787123
10590493149280566870405374771203135283200120394874883121348608054358179840611529523204026531840011811160064214573239881
1117714710392624207852480221709312014780620800662171811842060090081284484549836806726824755206643777536003897682821121030792151042567805035499
125314413401222474826892888683724806651279360034054550323212360540487683228875882496605414227968079725330432007015829078016371085174374489335319756830528348755377
131594323110539728265295347234586652672288222105600166015932825668865868431362098769323622447222309781504777321971712009120577801420872361609003008348407747051527696581394432360919263322915
144782969357128352928533715213205812838412105328435207747410198528361545809264641259261594173443305561684705286529504562380809576606691491841013062526042112731656268808192323256418566144659706976665604246293593703513
Technical details
Node id
n-4c9689f56a7176fd · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-9bf1c4f21fbefc3a

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:     3,     16;
  2:     9,     96,      192;
  3:    27,    432,     1728,      2048;
  4:    81,   1728,    10368,     24576,     20480;
  5:   243,   6480,    51840,    184320,    307200,     196608;
  6:   729,  23328,   233280,   1105920,   2764800,    3538944,    1835008;
  7:  2187,  81648,   979776,   5806080,  19353600,   37158912,   38535168,   16777216;
  8:  6561, 279936,  3919104,  27869184, 116121600,  297271296,  462422016,  402653184,  150994944;
  9: 19683, 944784, 15116544, 125411328, 627056640, 2006581248, 4161798144, 5435817984, 4076863488, 1342177280;
%F Formula
The row polynomials satisfy (3+8*n*x)*P(n,x) = (9+(24*n+48)*x+(64*n+64)*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[((9 + (24*n + 48)*x + (64*n + 64)*x^2)*P[n - 1, x])/(3 + 8*n*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] = ((9 + (24*n + 48)*x + (64*n + 64)*x^2)*v[i - 1])/(3 + 8*n*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

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.