Real-rooted atlas

A120910 → Veronese 2 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(48 - 48\,n - 2\,n x + 2\,n^{3} x\right)\,P_{n}(x) = \left(96 + 48\,x - 2\,x^{2} - 96\,n - 52\,n x - n x^{2} + 2\,n^{2} x^{2} + 4\,n^{3} x + n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-192\,x + 64\,x^{2} - 4\,x^{3} + 24\,n x^{2} - 6\,n x^{3} + 8\,n^{2} x^{2} - 2\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 3\)
\(P_{1}(x) = 6\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
033
166
212618
3243660
4481449201
59648090666
61921440540122184
7384403225201687104
8768107521008013441522959
915362764836288806427073806
1030726912012096040320270018236190
11614416896038016017740819800396752868
121228840550411404807096321188004752212391477
132457695846432947202635776617760411845467573026
1449152223641692252169225216288288028828876442423914836
Technical details
Node id
n-8171ffa220e5c9a0 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-0e03fcf0cb29044a

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:    3;
  1:    6;
  2:   12,     6;
  3:   24,    36;
  4:   48,   144,     9;
  5:   96,   480,    90;
  6:  192,  1440,   540,   12;
  7:  384,  4032,  2520,  168;
  8:  768, 10752, 10080, 1344,  15;
  9: 1536, 27648, 36288, 8064, 270;
%F Formula
The row polynomials satisfy -(48*n-48-(2*n^3-2*n)*x)*P(n,x) = -(96*n-96-(4*n^3-52*n+48)*x-(n^3+2*n^2-n-2)*x^2)*P(n-1,x) - (192*x-(8*n^2+24*n+64)*x^2+(2*n^2+6*n+4)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 3, P(1,x) = 6.
%t Mathematica
P[0, x_] := 3;
P[1, x_] := 6;
P[n_, x_] := P[n, x] = Expand[Together[(-(96*n - 96 - (4*n^3 - 52*n + 48)*x - (n^3 + 2*n^2 - n - 2)*x^2)*P[n - 1, x] - (192*x - (8*n^2 + 24*n + 64)*x^2 + (2*n^2 + 6*n + 4)*x^3)*D[P[n - 1, x], x])/(-48*n + 48 + (2*n^3 - 2*n)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 3; v[2] = 6; for(n = 2, N, my(i = n + 1); v[i] = (-(96*n - 96 - (4*n^3 - 52*n + 48)*x - (n^3 + 2*n^2 - n - 2)*x^2)*v[i - 1] - (192*x - (8*n^2 + 24*n + 64)*x^2 + (2*n^2 + 6*n + 4)*x^3)*deriv(v[i - 1], x))/(-48*n + 48 + (2*n^3 - 2*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.