Real-rooted atlas

A159854 → Veronese 2 (j=1) → D

nonnegative coefficients

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

Recurrence

\(\left(2 - 2\,n + n x - 2\,n^{2} x + n^{3} x\right)\,P_{n}(x) = \left(8\,x - 2\,n - 4\,n x - 2\,n x^{2} - n^{2} x - n^{2} x^{2} + n^{3} x + n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-4\,x + 4\,x^{2} + 2\,n x^{2} - 2\,n x^{3} + 2\,n^{2} x^{2} - 2\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
111
222
3325
44812
5520328
66401864
7770634144
8811216832320
991683781445704
1010240756480501536
11113301386132027563328
1212440237631681100727168
1313572386168643575468715360
14147286006137281001021849832768
Technical details
Node id
n-478a0e862ab5a96b · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-49176c5aecdbb25b

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): 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: 1;
  2: 2;
  3: 3,   2;
  4: 4,   8;
  5: 5,  20,   3;
  6: 6,  40,  18;
  7: 7,  70,  63,   4;
  8: 8, 112, 168,  32;
  9: 9, 168, 378, 144, 5;
%F Formula
The row polynomials satisfy -(2*n-2-(n^3-2*n^2+n)*x)*P(n,x) = -(2*n-(n^3-n^2-4*n+8)*x-(n^3-n^2-2*n)*x^2)*P(n-1,x) - (4*x-(2*n^2+2*n+4)*x^2+(2*n^2+2*n)*x^3)*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[(-(2*n - (n^3 - n^2 - 4*n + 8)*x - (n^3 - n^2 - 2*n)*x^2)*P[n - 1, x] - (4*x - (2*n^2 + 2*n + 4)*x^2 + (2*n^2 + 2*n)*x^3)*D[P[n - 1, x], x])/(-2*n + 2 + (n^3 - 2*n^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] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n - (n^3 - n^2 - 4*n + 8)*x - (n^3 - n^2 - 2*n)*x^2)*v[i - 1] - (4*x - (2*n^2 + 2*n + 4)*x^2 + (2*n^2 + 2*n)*x^3)*deriv(v[i - 1], x))/(-2*n + 2 + (n^3 - 2*n^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.