Real-rooted atlas

A007318 → Pₙ + Pₙ₊₁

nonnegative coefficients only rational roots

Proved for all n: all roots real what this means

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

Rows ?

nx0x1x2x3x4x5x6x7x8Pn(1)
0213
12316
2254112
32795124
42916146148
52112530207196
62133655502781192
72154991105773591384

Show all 15 stored rows

Generated b-files: A007283 (row sums) ?

Computed in your browser ?

Technical details
Node id
n-86fcb9f018178f39 · revision 1 · JSON
Definition source
derived:obreschkoff_combination
Fingerprint
f6-efce8cd29f4f1fd0

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (affine: slope 1, intercept 1)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • 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: 2,  1;
  1: 2,  3,  1;
  2: 2,  5,  4,   1;
  3: 2,  7,  9,   5,   1;
  4: 2,  9, 16,  14,   6,   1;
  5: 2, 11, 25,  30,  20,   7,   1;
  6: 2, 13, 36,  55,  50,  27,   8,   1;
  7: 2, 15, 49,  91, 105,  77,  35,   9,  1;
  8: 2, 17, 64, 140, 196, 182, 112,  44, 10,  1;
  9: 2, 19, 81, 204, 336, 378, 294, 156, 54, 11, 1;
%F Formula
The row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) for n >= 1, with P(0,x) = 2 + x.
%t Mathematica
P[0, x_] := 2 + x;
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] = 2 + x; 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.

Proved properties

  • Real-rooted. Every row P_n, n >= 0, 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-c4dc09241b591172

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

Same fingerprint (1)

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
  4. [Fis06] Steve Fisk. Polynomials, roots, and interlacing. arXiv:math/0612833. arXiv
  5. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  6. [Obr63] Nikola Obreschkoff. Verteilung und berechnung der nullstellen reeller polynome. Hochschulbücher für mathematik, VEB Deutscher Verlag der Wissenschaften, 1963.