Real-rooted atlas

basis-map coefficients of A113214, shifted → Veronese 2 (j=1) → D+1

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

Recurrence

\(\left(-12 + 18\,x - 4\,x^{2} + 17\,n - 22\,n x + 4\,n x^{2} - 6\,n^{2} + 4\,n^{2} x + n^{3}\right)\,P_{n}(x) = \left(12\,x - 28\,x^{2} + 8\,x^{3} - 8\,n + 2\,n x + 18\,n x^{2} - 4\,n x^{3} + 3\,n^{2} + 3\,n^{2} x - 4\,n^{2} x^{2} - n^{3} - n^{3} x\right)\,P_{n-1}(x) + \left(-8\,x + 28\,x^{2} - 28\,x^{3} + 8\,x^{4} + 6\,n x - 14\,n x^{2} + 8\,n x^{3} - 2\,n^{2} x + 2\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = -1\)

Rows ?

nx0x1x2x3Pn(1)
00
1-1-1
222
3-4-1-5
48412
5-15-12-1-28
62632664
7-42-77-24-1-144

All 15 rows

Technical details
Node id
n-d329df9122bfbea4 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-efd07601c6f432c5

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): no
  • 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:  -1;
  2:   2;
  3:  -4,   -1;
  4:   8,    4;
  5: -15,  -12,   -1;
  6:  26,   32,    6;
  7: -42,  -77,  -24,  -1;
  8:  64,  168,   80,   8;
  9: -93, -336, -234, -40, -1;
%F Formula
The row polynomials satisfy (n^3-6*n^2+17*n-12+(4*n^2-22*n+18)*x+(4*n-4)*x^2)*P(n,x) = -(n^3-3*n^2+8*n+(n^3-3*n^2-2*n-12)*x+(4*n^2-18*n+28)*x^2+(4*n-8)*x^3)*P(n-1,x) - ((2*n^2-6*n+8)*x-(2*n^2-14*n+28)*x^2-(8*n-28)*x^3-8*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = -1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := -1;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - 3*n^2 + 8*n + (n^3 - 3*n^2 - 2*n - 12)*x + (4*n^2 - 18*n + 28)*x^2 + (4*n - 8)*x^3)*P[n - 1, x] - ((2*n^2 - 6*n + 8)*x - (2*n^2 - 14*n + 28)*x^2 - (8*n - 28)*x^3 - 8*x^4)*D[P[n - 1, x], x])/(n^3 - 6*n^2 + 17*n - 12 + (4*n^2 - 22*n + 18)*x + (4*n - 4)*x^2)]];
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] = -1; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 - 3*n^2 + 8*n + (n^3 - 3*n^2 - 2*n - 12)*x + (4*n^2 - 18*n + 28)*x^2 + (4*n - 8)*x^3)*v[i - 1] - ((2*n^2 - 6*n + 8)*x - (2*n^2 - 14*n + 28)*x^2 - (8*n - 28)*x^3 - 8*x^4)*deriv(v[i - 1], x))/(n^3 - 6*n^2 + 17*n - 12 + (4*n^2 - 22*n + 18)*x + (4*n - 4)*x^2)); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.