Real-rooted atlas

A038220 → Veronese 2 (j=1) → θ+1

nonnegative coefficients

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

Recurrence

\(\left(27 - 27\,n + 6\,n x - 12\,n^{2} x + 6\,n^{3} x\right)\,P_{n}(x) = \left(144\,x - 81\,n - 72\,n x - 16\,n x^{2} - 18\,n^{2} x - 8\,n^{2} x^{2} + 18\,n^{3} x + 8\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-162\,x + 72\,x^{2} + 36\,n x^{2} - 16\,n x^{3} + 36\,n^{2} x^{2} - 16\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3Pn(1)
00
122
21212
3541670
4216192408
58101440962346
629168640172813284
710206453601814451274222

All 15 rows

Technical details
Node id
n-c301d8e5fa462de3 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-e77e0ae249f4ed07

Observed on the stored rows ?

  • Degrees: -1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6 (half affine: slope 1, intercept -1)
  • 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:      0;
  1:      2;
  2:     12;
  3:     54,     16;
  4:    216,    192;
  5:    810,   1440,     96;
  6:   2916,   8640,   1728;
  7:  10206,  45360,  18144,    512;
  8:  34992, 217728, 145152,  12288;
  9: 118098, 979776, 979776, 165888, 2560;
%F Formula
The row polynomials satisfy -(27*n-27-(6*n^3-12*n^2+6*n)*x)*P(n,x) = -(81*n-(18*n^3-18*n^2-72*n+144)*x-(8*n^3-8*n^2-16*n)*x^2)*P(n-1,x) - (162*x-(36*n^2+36*n+72)*x^2+(16*n^2+16*n)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-(81*n - (18*n^3 - 18*n^2 - 72*n + 144)*x - (8*n^3 - 8*n^2 - 16*n)*x^2)*P[n - 1, x] - (162*x - (36*n^2 + 36*n + 72)*x^2 + (16*n^2 + 16*n)*x^3)*D[P[n - 1, x], x])/(-27*n + 27 + (6*n^3 - 12*n^2 + 6*n)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = (-(81*n - (18*n^3 - 18*n^2 - 72*n + 144)*x - (8*n^3 - 8*n^2 - 16*n)*x^2)*v[i - 1] - (162*x - (36*n^2 + 36*n + 72)*x^2 + (16*n^2 + 16*n)*x^3)*deriv(v[i - 1], x))/(-27*n + 27 + (6*n^3 - 12*n^2 + 6*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.