Real-rooted atlas

A013610 → basis A008288

nonnegative coefficients

Proved for all n: all roots real, in [−3−2√2, −3+2√2] · coefficients of one sign · Hadamard kernel what this means

Recurrence
\(P_{n}(x) = \left(-4 + 6\,x\right)\,P_{n-2}(x) + \left(5 + 3\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 3\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1437
21633958
36422516227478
425612811647675813940
510246657127809180259224332476
64096327698496991422442269477729267688
7163841556495119747519775401081946433353421872206456

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-bd22381760780feb · revision 2 · JSON
Definition source
derived:basis_map_A008288
Fingerprint
f6-52999a0df2a941cc

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): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

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:      4,       3;
  2:     16,      33,        9;
  3:     64,     225,      162,       27;
  4:    256,    1281,     1647,      675,       81;
  5:   1024,    6657,    12780,     9180,     2592,      243;
  6:   4096,   32769,    84969,    91422,    44226,     9477,      729;
  7:  16384,  155649,   511974,   751977,   540108,   194643,    33534,    2187;
  8:  65536,  720897,  2883555,  5439933,  5328099,  2820987,   805545,  115911,   6561;
  9: 262144, 3276801, 15466464, 35914266, 45311724, 32551308, 13524408, 3188646, 393660, 19683;
%F Formula
The row polynomials satisfy P(n,x) = -(4-6*x)*P(n-2,x) + (5+3*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 3*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 3*x;
P[n_, x_] := P[n, x] = Expand[-(4 - 6*x)*P[n - 2, x] + (5 + 3*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; v[2] = 4 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = -(4 - 6*x)*v[i - 2] + (5 + 3*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 [-1564556493/268435456, -46056243/268435456]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-886ef664241d5f98
  • 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-1e942afebbc1521d, e-886ef664241d5f98
  • 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-1e942afebbc1521d, e-886ef664241d5f98
  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in [−3−2√2, −3+2√2] (rational outer bound [-1564556493/268435456, -46056243/268435456])
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-1e942afebbc1521d

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

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. [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