Real-rooted atlas

A132013: T(n,j) for an iterated mixed order Laguerre transform. Coefficients of the normalized generalized Laguerre polynomials (-1)^n*n!*L(n,1-n,x).

A023531 → (−1/n)D²+(1/n)D

A132013 only rational roots

Proved for all n: all roots real what this means

Recurrence from OEIS A132013
\(P_{n}(x) = \left(2\,x\right)\,P_{n-1}(x) + \left(-x^{2}\right)\,P_{n-2}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = -1 + x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2-110
30-21-1
400-31-2
5000-41-3
60000-51-4
700000-61-5
8000000-71-6
90000000-81-7
1000000000-91-8
11000000000-101-9
120000000000-111-10
1300000000000-121-11
14000000000000-131-12
150000000000000-141-13

Generated b-files: A001478 (row sums), A001489 (row sums), A024000 (row sums), A132013 ?

Computed in your browser ?

Technical details
Node id
n-ae37fdb1a61214fb · revision 1 · JSON
Definition source
oeis_catalog:A132013
Fingerprint
f6-e61870f548717ba4

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept -1)
  • 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): yes

OEIS matches ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:  1;
   2: -1,  1;
   3:  0, -2,  1;
   4:  0,  0, -3,  1;
   5:  0,  0,  0, -4,  1;
   6:  0,  0,  0,  0, -5,  1;
   7:  0,  0,  0,  0,  0, -6,  1;
   8:  0,  0,  0,  0,  0,  0, -7,  1;
   9:  0,  0,  0,  0,  0,  0,  0, -8,  1;
  10:  0,  0,  0,  0,  0,  0,  0,  0, -9, 1;
%F Formula
The row polynomials satisfy P(n,x) = 2*x*P(n-1,x) - x^2*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = -1 + x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := -1 + x;
P[n_, x_] := P[n, x] = Expand[2*x*P[n - 1, x] - x^2*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = -1 + x; for(n = 3, N, my(i = n); v[i] = 2*x*v[i - 1] - x^2*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Generated with polytool c48629374597.

A132013: triangle, offset 0, from the row P1
%e Example
Triangle begins:
  0:  1;
  1: -1,  1;
  2:  0, -2,  1;
  3:  0,  0, -3,  1;
  4:  0,  0,  0, -4,  1;
  5:  0,  0,  0,  0, -5,  1;
  6:  0,  0,  0,  0,  0, -6,  1;
  7:  0,  0,  0,  0,  0,  0, -7,  1;
  8:  0,  0,  0,  0,  0,  0,  0, -8,  1;
  9:  0,  0,  0,  0,  0,  0,  0,  0, -9, 1;
%F Formula
The row polynomials satisfy P(n,x) = 2*x*P(n-1,x) - x^2*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = -1 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := -1 + x;
P[n_, x_] := P[n, x] = Expand[2*x*P[n - 1, x] - x^2*P[n - 2, x]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = -1 + x; for(n = 2, N, my(i = n + 1); v[i] = 2*x*v[i - 1] - x^2*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-b403369ae2d1e52b

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

No candidate equalities or related sequences.

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. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.