Real-rooted atlas

A013611 → Veronese 2 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(3 - 3\,n - 8\,n x + 8\,n^{3} x\right)\,P_{n}(x) = \left(3 + 96\,x - 256\,x^{2} - 3\,n - 104\,n x - 128\,n x^{2} + 256\,n^{2} x^{2} + 8\,n^{3} x + 128\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-6\,x + 128\,x^{2} - 512\,x^{3} + 48\,n x^{2} - 768\,n x^{3} + 16\,n^{2} x^{2} - 256\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
111
213233
319697
41192768961
5132038404161
61480115201638428385
7167226880114688142241
8189653760458752327680841089
91115296768137625629491204423297
1011440161280344064014745600629145624640417
111176025344075694085406720069206016131097825
121211238016015138816162201600415236096117440512710399297
131249654912028114944421724160179935641615267266563776473793
141291276876849201152984023040629774745610687086592214748364820166313569
Technical details
Node id
n-464ea3b0fca68ea4 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-62e9f29822ab661c

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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: 1;
  2: 1,   32;
  3: 1,   96;
  4: 1,  192,   768;
  5: 1,  320,  3840;
  6: 1,  480, 11520,   16384;
  7: 1,  672, 26880,  114688;
  8: 1,  896, 53760,  458752,  327680;
  9: 1, 1152, 96768, 1376256, 2949120;
%F Formula
The row polynomials satisfy -(3*n-3-(8*n^3-8*n)*x)*P(n,x) = -(3*n-3-(8*n^3-104*n+96)*x-(128*n^3+256*n^2-128*n-256)*x^2)*P(n-1,x) - (6*x-(16*n^2+48*n+128)*x^2+(256*n^2+768*n+512)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(3*n - 3 - (8*n^3 - 104*n + 96)*x - (128*n^3 + 256*n^2 - 128*n - 256)*x^2)*P[n - 1, x] - (6*x - (16*n^2 + 48*n + 128)*x^2 + (256*n^2 + 768*n + 512)*x^3)*D[P[n - 1, x], x])/(-3*n + 3 + (8*n^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; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(3*n - 3 - (8*n^3 - 104*n + 96)*x - (128*n^3 + 256*n^2 - 128*n - 256)*x^2)*v[i - 1] - (6*x - (16*n^2 + 48*n + 128)*x^2 + (256*n^2 + 768*n + 512)*x^3)*deriv(v[i - 1], x))/(-3*n + 3 + (8*n^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

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.