Real-rooted atlas

A103633 → Veronese 2 → D

nonnegative coefficients

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

Recurrence

\(\left(4\right)\,P_{n}(x) = \left(-2 + 10\,x - 18\,x^{2} + 4\,x^{3}\right)\,P_{n-6}(x) + \left(-x + 5\,x^{2} - 4\,x^{3}\right)\,P_{n-6}'(x) + \left(-4 + 16\,x - 12\,x^{2}\right)\,P_{n-4}(x) + \left(-2\,x + 8\,x^{2}\right)\,P_{n-4}'(x) + \left(2 + 12\,x\right)\,P_{n-2}(x) + \left(1 - 4\,x\right)\,P_{n-2}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 1\)
\(P_{3}(x) = 1\)
\(P_{4}(x) = 1 + 2\,x\)
\(P_{5}(x) = 4\,x\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
10
211
311
4123
5044
60639
702911
80218424
900121628
10001540560
11003402568
1200360756144
130002410036160
14000281751267336
Technical details
Node id
n-89eea40781af2171 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-16c35582abfc142d

Observed on the stored rows ?

  • Degrees: -1, -1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6 (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;
  1: 0;
  2: 1;
  3: 1;
  4: 1, 2;
  5: 0, 4;
  6: 0, 6,  3;
  7: 0, 2,  9;
  8: 0, 2, 18,  4;
  9: 0, 0, 12, 16;
%F Formula
The row polynomials satisfy 4*P(n,x) = -(2-10*x+18*x^2-4*x^3)*P(n-6,x) - (x-5*x^2+4*x^3)*d/dx P(n-6,x) - (4-16*x+12*x^2)*P(n-4,x) - (2*x-8*x^2)*d/dx P(n-4,x) + (2+12*x)*P(n-2,x) + (1-4*x)*d/dx P(n-2,x) for n >= 6, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 1, P(3,x) = 1, P(4,x) = 1 + 2*x, P(5,x) = 4*x.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 1;
P[3, x_] := 1;
P[4, x_] := 1 + 2*x;
P[5, x_] := 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(2 - 10*x + 18*x^2 - 4*x^3)*P[n - 6, x] - (x - 5*x^2 + 4*x^3)*D[P[n - 6, x], x] - (4 - 16*x + 12*x^2)*P[n - 4, x] - (2*x - 8*x^2)*D[P[n - 4, x], x] + (2 + 12*x)*P[n - 2, x] + (1 - 4*x)*D[P[n - 2, x], x])/(4)]];
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] = 0; v[3] = 1; v[4] = 1; v[5] = 1 + 2*x; v[6] = 4*x; for(n = 6, N, my(i = n + 1); v[i] = (-(2 - 10*x + 18*x^2 - 4*x^3)*v[i - 6] - (x - 5*x^2 + 4*x^3)*deriv(v[i - 6], x) - (4 - 16*x + 12*x^2)*v[i - 4] - (2*x - 8*x^2)*deriv(v[i - 4], x) + (2 + 12*x)*v[i - 2] + (1 - 4*x)*deriv(v[i - 2], x))/(4)); 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.