Real-rooted atlas

A013622 → 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
\(P_{n}(x) = \left(-24 + 10\,x\right)\,P_{n-2}(x) + \left(11 + 5\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 8 + 5\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
18513
26410525194
351214358501252922
440961646516975587562544036
532768172275263000155000375003125663668
626214417046653511625304875012312502281251562510002184
72097152162551154256195050004375294375008965625134375078125150743592

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-55489dd386249f31 · revision 2 · JSON
Definition source
derived:basis_map_A008288
Fingerprint
f6-c76c75d4b81622de

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:         8,          5;
  2:        64,        105,         25;
  3:       512,       1435,        850,        125;
  4:      4096,      16465,      16975,       5875,        625;
  5:     32768,     172275,     263000,     155000,      37500,       3125;
  6:    262144,    1704665,    3511625,    3048750,    1231250,     228125,      15625;
  7:   2097152,   16255115,   42561950,   50004375,   29437500,    8965625,    1343750,     78125;
  8:  16777216,  151001505,  482224675,  724804125,  574771875,  252646875,   61515625,   7734375,   390625;
  9: 134217728, 1375751395, 5200543300, 9609483250, 9740055000, 5732175000, 1997312500, 404218750, 43750000, 1953125;
%F Formula
The row polynomials satisfy P(n,x) = -(24-10*x)*P(n-2,x) + (11+5*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 8 + 5*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 8 + 5*x;
P[n_, x_] := P[n, x] = Expand[-(24 - 10*x)*P[n - 2, x] + (11 + 5*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] = 8 + 5*x; for(n = 2, N, my(i = n + 1); v[i] = -(24 - 10*x)*v[i - 2] + (11 + 5*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-b0660dd698ef4dd1
  • 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-b0660dd698ef4dd1, e-e798b3029e523eaa
  • 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-b0660dd698ef4dd1, e-e798b3029e523eaa
  • 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-e798b3029e523eaa

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