Real-rooted atlas

A159854 → Veronese 2 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(20 - 6\,x - 20\,n + n x + 4\,n^{2} x + n^{3} x\right)\,P_{n}(x) = \left(60 - 40\,x + 12\,x^{2} - 20\,n - 43\,n x + 19\,n x^{2} + 2\,n^{2} x + 8\,n^{2} x^{2} + n^{3} x + n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-40\,x + 64\,x^{2} - 24\,x^{3} + 14\,n x^{2} - 14\,n x^{3} + 2\,n^{2} x^{2} - 2\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 2\,x\)
\(P_{1}(x) = 2\,x\)

Rows ?

nx0x1x2x3x4Pn(1)
0022
1022
20235
302911
40218424
502302052
60245605112
7026314035240

All 15 rows

Technical details
Node id
n-6fe628e730db8ca3 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-7ee7a9481aee855d

Observed on the stored rows ?

  • Degrees: 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8 (half affine: slope 1, intercept 2)
  • 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, 2;
  1: 0, 2;
  2: 0, 2,   3;
  3: 0, 2,   9;
  4: 0, 2,  18,   4;
  5: 0, 2,  30,  20;
  6: 0, 2,  45,  60,   5;
  7: 0, 2,  63, 140,  35;
  8: 0, 2,  84, 280, 140,  6;
  9: 0, 2, 108, 504, 420, 54;
%F Formula
The row polynomials satisfy -(20*n-20-(n^3+4*n^2+n-6)*x)*P(n,x) = -(20*n-60-(n^3+2*n^2-43*n-40)*x-(n^3+8*n^2+19*n+12)*x^2)*P(n-1,x) - (40*x-(2*n^2+14*n+64)*x^2+(2*n^2+14*n+24)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 2*x, P(1,x) = 2*x.
%t Mathematica
P[0, x_] := 2*x;
P[1, x_] := 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(20*n - 60 - (n^3 + 2*n^2 - 43*n - 40)*x - (n^3 + 8*n^2 + 19*n + 12)*x^2)*P[n - 1, x] - (40*x - (2*n^2 + 14*n + 64)*x^2 + (2*n^2 + 14*n + 24)*x^3)*D[P[n - 1, x], x])/(-20*n + 20 + (n^3 + 4*n^2 + n - 6)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 2*x; v[2] = 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(20*n - 60 - (n^3 + 2*n^2 - 43*n - 40)*x - (n^3 + 8*n^2 + 19*n + 12)*x^2)*v[i - 1] - (40*x - (2*n^2 + 14*n + 64)*x^2 + (2*n^2 + 14*n + 24)*x^3)*deriv(v[i - 1], x))/(-20*n + 20 + (n^3 + 4*n^2 + n - 6)*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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.