Real-rooted atlas

A033282 → basis A013611

nonnegative coefficients

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · interlacing (weak) · Hadamard kernel what this means

Recurrence
\(\left(2 + n\right)\,P_{n}(x) = \left(1 - n\right)\,P_{n-2}(x) + \left(3 + 8\,x + 6\,n + 16\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 8\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
13811
2116080151
34537210088962321
419721848960161281075238221
59031256069120188160253440135168659351
6427971604494640180576036748803953664175718411762011
72079340670033818401549856042282240686766086150144023429120215197301
810304923068322243225612377164842384742492252160012464291849559080963186360324015956121
951885913081752145656576940063488387584870410587193344191616122882215968768014860025856440297062476146659171
10264672374213100931017360688169664033191208960109152755712247884349440383884001280388011786240231155957760616415887361462811221951
111364886942129872458799109604898027904027058977792010409885122562845885464576552900919296074814514790406715376926720359892045004887246556364828409982504761
12710393732393631000367865625603409860326402123227560960935652954931229922313175040699735755980801187601835622401426714892697601151654544015365608707194880012463793766400556903876097701
133726935191361153539222841179545623318668538881615915587686480266945634304294037808283648804315410595840164293931866521624752003838115842673483762892800196055475945472087495832240128017947863023616011003968760730671
14196880151977471954532140946731889615714445900608119952357068544663157096842240273838941478502485801597350543362049956334973747237165601622589440503339015942963204938046032052224033175503057715200136628107267276802602440138424320218939142767738371

Computed in your browser ?

Technical details
Node id
n-d1ffe586e8c0e96c · revision 1 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-9581fdced757748c

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:      3,        8;
  2:     11,       60,        80;
  3:     45,      372,      1008,       896;
  4:    197,     2184,      8960,     16128,      10752;
  5:    903,    12560,     69120,    188160,     253440,      135168;
  6:   4279,    71604,    494640,   1805760,    3674880,     3953664,     1757184;
  7:  20793,   406700,   3381840,  15498560,   42282240,    68676608,    61501440,    23429120;
  8: 103049,  2306832,  22432256, 123771648,  423847424,   922521600,  1246429184,   955908096,   318636032;
  9: 518859, 13081752, 145656576, 940063488, 3875848704, 10587193344, 19161612288, 22159687680, 14860025856, 4402970624;
%F Formula
The row polynomials satisfy (n+2)*P(n,x) = -(n-1)*P(n-2,x) + (6*n+3+(16*n+8)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 8*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n - 1)*P[n - 2, x] + (6*n + 3 + (16*n + 8)*x)*P[n - 1, x])/(n + 2)]];
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] = 3 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n - 1)*v[i - 2] + (6*n + 3 + (16*n + 8)*x)*v[i - 1])/(n + 2)); 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-ca122abf64af43e3
  • 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-ca122abf64af43e3
  • 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-ca122abf64af43e3
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-ca122abf64af43e3

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

No candidate equalities or related sequences.

References

  1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023