Real-rooted atlas

A159834: Coefficient array of Hermite_H(n, (x-1)/sqrt(2))/(sqrt(2))^n.

A066325 → x ↦ x−1 (shift -1)

A159834

Proved for all n: all roots real · interlacing (weak) what this means

Recurrence from OEIS A159834
\(P_{n}(x) = \left(-1 + x\right)\,P_{n-1}(x) + \left(-1\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2-110
30-21-1
420-310
5-280-413
6-6-10200-510
716-36-30400-61-15
820112-126-70700-710

Show all 15 stored rows

Generated b-files: A159834 ?

Computed in your browser ?

Technical details
Node id
n-f0c4b6406e0da4a3 · revision 1 · JSON
Definition source
oeis_catalog:A159834
Fingerprint
f6-734d2c9bd67beec2

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): no

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:    2,     0,   -3,    1;
   5:   -2,     8,    0,   -4,    1;
   6:   -6,   -10,   20,    0,   -5,    1;
   7:   16,   -36,  -30,   40,    0,   -6,   1;
   8:   20,   112, -126,  -70,   70,    0,  -7,  1;
   9: -132,   160,  448, -336, -140,  112,   0, -8,  1;
  10:  -28, -1188,  720, 1344, -756, -252, 168,  0, -9, 1;
%F Formula
The row polynomials satisfy P(n,x) = -(1-x)*P(n-1,x) - d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(1 - x)*P[n - 1, x] - D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = -(1 - x)*v[i - 1] - deriv(v[i - 1], x)); 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.

A159834: triangle, offset 0, from the row P1
%e Example
Triangle begins:
  0:    1;
  1:   -1,     1;
  2:    0,    -2,    1;
  3:    2,     0,   -3,    1;
  4:   -2,     8,    0,   -4,    1;
  5:   -6,   -10,   20,    0,   -5,    1;
  6:   16,   -36,  -30,   40,    0,   -6,   1;
  7:   20,   112, -126,  -70,   70,    0,  -7,  1;
  8: -132,   160,  448, -336, -140,  112,   0, -8,  1;
  9:  -28, -1188,  720, 1344, -756, -252, 168,  0, -9, 1;
%F Formula
The row polynomials satisfy P(n,x) = -(1-x)*P(n-1,x) - d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(1 - x)*P[n - 1, x] - D[P[n - 1, x], 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; for(n = 1, N, my(i = n + 1); v[i] = -(1 - x)*v[i - 1] - deriv(v[i - 1], x)); 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-9c6e37fd70f39446, e-c585175918782f17
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-9c6e37fd70f39446

Proof chains

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

Used in 2 proof steps

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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  4. [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
  5. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  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.
  8. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023