Real-rooted atlas

basis-map coefficients of A113214, shifted → Veronese 2 → x ↦ x+1

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1-1-1
2213
3-4-3-7
488117
5-16-20-5-41
6324818199
7-64-112-56-7-239
8128256160321577
9-256-576-432-120-9-1393
10512128011204005013363
11-1024-2816-2816-1232-220-11-8119
12204861446912358484072119601
13-4096-13312-16640-9984-2912-364-13-47321
14819228672394242688094081568981114243

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(P_{n}(x) = \left(x\right)\,P_{n-2}(x) + \left(-2\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = -1\)
Technical details
Node id
n-777834828b7583c2 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-51eae1c415038b42)
Fingerprint
f6-702b8764f2c773f5

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,    8,    1;
  5:  -16,  -20,   -5;
  6:   32,   48,   18,    1;
  7:  -64, -112,  -56,   -7;
  8:  128,  256,  160,   32,  1;
  9: -256, -576, -432, -120, -9;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(2*n-2+(n-1)*x)*P(n-1,x) + (2*x+2*x^2)*d/dx 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[Together[(-(2*n - 2 + (n - 1)*x)*P[n - 1, x] + (2*x + 2*x^2)*D[P[n - 1, x], x])/(n - 1)]];
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] = (-(2*n - 2 + (n - 1)*x)*v[i - 1] + (2*x + 2*x^2)*deriv(v[i - 1], x))/(n - 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

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.