Real-rooted atlas

A013611 → 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
\(P_{n}(x) = \left(-5 + 12\,x\right)\,P_{n-2}(x) + \left(6 + 4\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 5 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1549
225561697
3125476368641033
462533764704204825611009
53125218764560035200104961024117321
6156251343763780964398082270725120040961250273
77812579687628405924522432349158413363202416641638413323977

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-215b7d121fbc1e4b · revision 2 · JSON
Definition source
derived:basis_map_A008288
Fingerprint
f6-32550b608f1e0aa0

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:       5,        4;
  2:      25,       56,        16;
  3:     125,      476,       368,        64;
  4:     625,     3376,      4704,      2048,       256;
  5:    3125,    21876,     45600,     35200,     10496,      1024;
  6:   15625,   134376,    378096,    439808,    227072,     51200,      4096;
  7:   78125,   796876,   2840592,   4522432,   3491584,   1336320,    241664,    16384;
  8:  390625,  4609376,  19953088,  40835072,  43181568,  24453120,   7389184,  1114112,   65536;
  9: 1953125, 26171876, 133515584, 336297728, 459240960, 354662400, 156975104, 39059456, 5046272, 262144;
%F Formula
The row polynomials satisfy P(n,x) = -(5-12*x)*P(n-2,x) + (6+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 5 + 4*x;
P[n_, x_] := P[n, x] = Expand[-(5 - 12*x)*P[n - 2, x] + (6 + 4*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] = 5 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(5 - 12*x)*v[i - 2] + (6 + 4*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-1d35ea8dc489f8c6
  • 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-1d35ea8dc489f8c6, e-cfd0ddf0ef7532e7
  • 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-1d35ea8dc489f8c6, e-cfd0ddf0ef7532e7
  • 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-cfd0ddf0ef7532e7

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