Real-rooted atlas

Veronese parent of A109446 → Veronese 2 (j=1) → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-13 + 27\,x - 15\,x^{2} + x^{3}\right)\,P_{n-6}(x) + \left(4 - 12\,x + 12\,x^{2} - 4\,x^{3}\right)\,P_{n-6}'(x) + \left(-23 + 10\,x - 3\,x^{2}\right)\,P_{n-4}(x) + \left(-8 + 8\,x^{2}\right)\,P_{n-4}'(x) + \left(5 + 3\,x\right)\,P_{n-2}(x) + \left(4 - 4\,x\right)\,P_{n-2}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1 + 2\,x\)
\(P_{3}(x) = 3 + 2\,x\)
\(P_{4}(x) = 1 + 12\,x + 3\,x^{2}\)
\(P_{5}(x) = 5 + 20\,x + 3\,x^{2}\)

Rows ?

nx0x1x2x3Pn(1)
011
111
2123
3325
4112316
5520328
613045480
7770634144

All 15 rows

Technical details
Node id
n-2ccaa62fe6a23d72 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-f47000076ec3302d

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: 1;
  1: 1;
  2: 1,   2;
  3: 3,   2;
  4: 1,  12,   3;
  5: 5,  20,   3;
  6: 1,  30,  45,   4;
  7: 7,  70,  63,   4;
  8: 1,  56, 210, 112, 5;
  9: 9, 168, 378, 144, 5;
%F Formula
The row polynomials satisfy P(n,x) = -(13-27*x+15*x^2-x^3)*P(n-6,x) + (4-12*x+12*x^2-4*x^3)*d/dx P(n-6,x) - (23-10*x+3*x^2)*P(n-4,x) - (8-8*x^2)*d/dx P(n-4,x) + (5+3*x)*P(n-2,x) + (4-4*x)*d/dx P(n-2,x) for n >= 6, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 1 + 2*x, P(3,x) = 3 + 2*x, P(4,x) = 1 + 12*x + 3*x^2, P(5,x) = 5 + 20*x + 3*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1;
P[2, x_] := 1 + 2*x;
P[3, x_] := 3 + 2*x;
P[4, x_] := 1 + 12*x + 3*x^2;
P[5, x_] := 5 + 20*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[-(13 - 27*x + 15*x^2 - x^3)*P[n - 6, x] + (4 - 12*x + 12*x^2 - 4*x^3)*D[P[n - 6, x], x] - (23 - 10*x + 3*x^2)*P[n - 4, x] - (8 - 8*x^2)*D[P[n - 4, x], x] + (5 + 3*x)*P[n - 2, x] + (4 - 4*x)*D[P[n - 2, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; v[3] = 1 + 2*x; v[4] = 3 + 2*x; v[5] = 1 + 12*x + 3*x^2; v[6] = 5 + 20*x + 3*x^2; for(n = 6, N, my(i = n + 1); v[i] = -(13 - 27*x + 15*x^2 - x^3)*v[i - 6] + (4 - 12*x + 12*x^2 - 4*x^3)*deriv(v[i - 6], x) - (23 - 10*x + 3*x^2)*v[i - 4] - (8 - 8*x^2)*deriv(v[i - 4], x) + (5 + 3*x)*v[i - 2] + (4 - 4*x)*deriv(v[i - 2], 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.