Real-rooted atlas

A023531: a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.

A023531 palindromic γ-positive nonnegative coefficients only rational roots

Proved for all n: all roots real, in [0, ∞) · all roots real, in [−1, 0] · coefficients of one sign · alternating coefficient signs · Hadamard kernel basis map preserves real roots what this means

Recurrence from OEIS A023531
\(P_{n}(x) = \left(x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1011
20011
300011
4000011
50000011
600000011
7000000011

Show all 15 stored rows

Computed in your browser ?

Other recurrences

X ↦ x/(x+1) substitution · proved equal for all n (proof step)

\(\left(1 + x\right)\,P_{n}(x) = \left(x + x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
Technical details
Node id
n-6fea5aff4f44fa54 · revision 4 · JSON
Definition source
oeis_catalog:A023531
derived:mobius_x_over_x_plus_one (equal by e-87ce49be7a0dde6c)
Fingerprint
f6-edf6954a4cd8adfc

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: yes
  • γ-positive (gamma_positive): yes
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): yes

OEIS matches ?

No OEIS matches.

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: 0, 1;
  2: 0, 0, 1;
  3: 0, 0, 0, 1;
  4: 0, 0, 0, 0, 1;
  5: 0, 0, 0, 0, 0, 1;
  6: 0, 0, 0, 0, 0, 0, 1;
  7: 0, 0, 0, 0, 0, 0, 0, 1;
  8: 0, 0, 0, 0, 0, 0, 0, 0, 1;
  9: 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
%F Formula
The row polynomials satisfy P(n,x) = 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[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] = 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.

Proved properties

  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-292206ad4ae86d2d, e-b7eca039219a7f53
  • Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n 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-292206ad4ae86d2d, e-7e44b66e04c497d6, e-b7eca039219a7f53
  • Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
    Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
    derived from the cited steps by a stated theorem: e-292206ad4ae86d2d, e-7e44b66e04c497d6, e-b7eca039219a7f53
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, +inf) to roots in (-inf, +inf): every real-rooted input has a real-rooted image (an affine substitution times g)
    The linear map sending x^k to the k-th polynomial of the sequence turns polynomials with roots in one interval into real-rooted ones.
    by the cited steps: e-d4a7bd772f22ef34
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, 0] to roots in (-inf, 0]: an input root r <= 0 gives the root (r - a)/d <= 0; a larger root gives a positive root
    The linear map sending x^k to the k-th polynomial of the sequence turns polynomials with roots in one interval into real-rooted ones.
    by the cited steps: e-d4a7bd772f22ef34
  • 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-61d6f3dd155b69fd, e-7d893ef04d3ac08a, e-82c705261aa969bd, e-f913bfdfa15279fe
  • 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-61d6f3dd155b69fd, e-7d893ef04d3ac08a, e-82c705261aa969bd, e-f913bfdfa15279fe
  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-7e44b66e04c497d6

Proof chains

9 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 32 proof steps

Same fingerprint (75)

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