Real-rooted atlas

A023531 → x ↦ x−1

only rational roots

Proved for all n: all roots real, in (−∞, 1] · all roots real, in [0, ∞) · alternating coefficient signs what this means

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1-110
21-210
3-13-310
41-46-410
5-15-1010-510
61-615-2015-610
7-17-2135-3521-710
81-828-5670-5628-810
9-19-3684-126126-8436-910
101-1045-120210-252210-12045-1010
11-111-55165-330462-462330-16555-1110
121-1266-220495-792924-792495-22066-1210
13-113-78286-7151287-17161716-1287715-28678-1310
141-1491-3641001-20023003-34323003-20021001-36491-1410

Generated b-files: A130595 ?

Computed in your browser ?

Technical details
Node id
n-50902364601606aa · revision 2 · JSON
Definition source
derived:basis_map_coefficients_of:oeis_catalog:A048993
Fingerprint
f6-85f6fce8b02c5e6d

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 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): 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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:  1;
  1: -1,  1;
  2:  1, -2,   1;
  3: -1,  3,  -3,   1;
  4:  1, -4,   6,  -4,    1;
  5: -1,  5, -10,  10,   -5,   1;
  6:  1, -6,  15, -20,   15,  -6,   1;
  7: -1,  7, -21,  35,  -35,  21,  -7,  1;
  8:  1, -8,  28, -56,   70, -56,  28, -8,  1;
  9: -1,  9, -36,  84, -126, 126, -84, 36, -9, 1;
%F Formula
The row polynomials satisfy P(n,x) = -(1-x)*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]];
Table[CoefficientList[P[n, x], x], {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]); 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)

Generated with polytool c48629374597.

A130595: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:  1;
  1: -1,  1;
  2:  1, -2,   1;
  3: -1,  3,  -3,   1;
  4:  1, -4,   6,  -4,    1;
  5: -1,  5, -10,  10,   -5,   1;
  6:  1, -6,  15, -20,   15,  -6,   1;
  7: -1,  7, -21,  35,  -35,  21,  -7,  1;
  8:  1, -8,  28, -56,   70, -56,  28, -8,  1;
  9: -1,  9, -36,  84, -126, 126, -84, 36, -9, 1;
%F Formula
The row polynomials satisfy P(n,x) = -(1-x)*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]];
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]); 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 >= 0, is zero or real-rooted with all roots in (-inf, 1]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-0ba23af98afc4615, e-8c1f314a0c6d10ae
  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in [0, +inf)
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-c8c4101b2f8f8c16
  • Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n(-x) have one sign
    All coefficients of each polynomial have the same sign (after x -> -x when alternating).
    derived from the cited steps by a stated theorem: e-c8c4101b2f8f8c16

Proof chains

3 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

Same fingerprint (2)

Equal up to content, sign, reversal and a small index shift.

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