Real-rooted atlas

basis-map coefficients of A113214, shifted → Veronese 2 → basis A008277

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

Recurrence

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

Rows ?

nx0x1x2x3Pn(1)
011
1-1-1
2213
3-4-3-7
489118
5-16-25-5-46
63267211121
7-64-175-77-7-323

All 15 rows

Technical details
Node id
n-6d9d7b3039e7167b · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-d79d0ea0a079bfa0

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): 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:    1;
  1:   -1;
  2:    2,     1;
  3:   -4,    -3;
  4:    8,     9,    1;
  5:  -16,   -25,   -5;
  6:   32,    67,   21,    1;
  7:  -64,  -175,  -77,   -7;
  8:  128,   449,  263,   38,  1;
  9: -256, -1137, -855, -174, -9;
%F Formula
The row polynomials satisfy P(n,x) = x*P(n-2,x) + x*d/dx P(n-2,x) - 2*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = -1.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := -1;
P[n_, x_] := P[n, x] = Expand[x*P[n - 2, x] + x*D[P[n - 2, x], x] - 2*P[n - 1, 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; for(n = 2, N, my(i = n + 1); v[i] = x*v[i - 2] + x*deriv(v[i - 2], x) - 2*v[i - 1]); 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

  • Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
    by Ray basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
    derived from Ray basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
    derived from Ray basis map theorem

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.