Real-rooted atlas

A145597 → basis A013610

nonnegative coefficients

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

Recurrence
\(\left(6\,n + n^{2}\right)\,P_{n}(x) = \left(-18\,x^{2} - 27\,n x^{2} - 9\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(20 + 30\,x + 18\,n + 27\,n x + 4\,n^{2} + 6\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 6 + 9\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
16915
2278154162
31104956752701550
4429257453464455121514019
51638122853439844226255155103123165
6618855692196560343980309582132678204121065092
7232562441881041012231336028576801928934642978787329130140

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-b388c59c06b1a9a4 · revision 1 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-15bfbb956e5929fa

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:      6,       9;
  2:     27,      81,       54;
  3:    110,     495,      675,      270;
  4:    429,    2574,     5346,     4455,      1215;
  5:   1638,   12285,    34398,    44226,     25515,      5103;
  6:   6188,   55692,   196560,   343980,    309582,    132678,     20412;
  7:  23256,  244188,  1041012,  2313360,   2857680,   1928934,    642978,    78732;
  8:  87210, 1046520,  5232600, 14128020,  22307400,  20820240,  11022480,  2952450,   295245;
  9: 326876, 4412826, 25325784, 80582040, 155408220, 186489864, 137413584, 58891536, 12990780, 1082565;
%F Formula
The row polynomials satisfy (n^2+6*n)*P(n,x) = -(9*n^2+27*n+18)*x^2*P(n-2,x) + (4*n^2+18*n+20+(6*n^2+27*n+30)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + 9*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 6 + 9*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(9*n^2 + 27*n + 18)*x^2*P[n - 2, x] + (4*n^2 + 18*n + 20 + (6*n^2 + 27*n + 30)*x)*P[n - 1, x])/(n^2 + 6*n)]];
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] = 6 + 9*x; for(n = 2, N, my(i = n + 1); v[i] = (-(9*n^2 + 27*n + 18)*x^2*v[i - 2] + (4*n^2 + 18*n + 20 + (6*n^2 + 27*n + 30)*x)*v[i - 1])/(n^2 + 6*n)); 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-d6186b9d95d911cd
  • 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-d6186b9d95d911cd
  • 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-d6186b9d95d911cd
  • 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-d6186b9d95d911cd

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