Real-rooted atlas

A172431 → basis A013609 → reverse

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(4\,n\right)\,P_{n}(x) = \left(8 + 14\,x + 5\,x^{2} + 8\,n - 2\,n x - 5\,n x^{2}\right)\,P_{n-1}(x) + \left(16\,x + 18\,x^{2} + 5\,x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1437
21220840
332887621217
480320444260551159
51921040204818968401446160
644831368176105927344261637732689
710248960295685008048160266647940987173383

All 15 rows

Technical details
Node id
n-1ebae075a566c573 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-ade2b5b96bd8d678

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

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:    1;
  1:    4,     3;
  2:   12,    20,      8;
  3:   32,    88,     76,     21;
  4:   80,   320,    444,    260,      55;
  5:  192,  1040,   2048,   1896,     840,     144;
  6:  448,  3136,   8176,  10592,    7344,    2616,    377;
  7: 1024,  8960,  29568,  50080,   48160,   26664,   7940,    987;
  8: 2304, 24576,  99456, 210816,  262960,  200704,  92436,  23644,  2584;
  9: 5120, 65280, 316416, 814080, 1261760, 1242320, 785280, 309512, 69380, 6765;
%F Formula
The row polynomials satisfy 4*n*P(n,x) = (8*n+8-(2*n-14)*x-(5*n-5)*x^2)*P(n-1,x) + (16*x+18*x^2+5*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((8*n + 8 - (2*n - 14)*x - (5*n - 5)*x^2)*P[n - 1, x] + (16*x + 18*x^2 + 5*x^3)*D[P[n - 1, x], x])/(4*n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((8*n + 8 - (2*n - 14)*x - (5*n - 5)*x^2)*v[i - 1] + (16*x + 18*x^2 + 5*x^3)*deriv(v[i - 1], x))/(4*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)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

References

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