Real-rooted atlas

A207823 → basis A013610 → D

nonnegative coefficients

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

Recurrence

\(\left(-2 + 2\,n\right)\,P_{n}(x) = \left(10 + 6\,x + 5\,n + 3\,n x\right)\,P_{n-1}(x) + \left(7 + 10\,x + 3\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
133
2301848
321927081570
41410264616203246000
5848421420199268100121559145
648930155358194400119880364504374559392
7274113104895016511851368900630585153090153095142132

All 15 rows

Technical details
Node id
n-4d1bce9bffc67060 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-3fb1e6fd5f3882b1

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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:       0;
  1:       3;
  2:      30,       18;
  3:     219,      270,       81;
  4:    1410,     2646,     1620,       324;
  5:    8484,    21420,    19926,      8100,     1215;
  6:   48930,   155358,   194400,    119880,    36450,     4374;
  7:  274113,  1048950,  1651185,   1368900,   630585,   153090,    15309;
  8: 1503450,  6734070, 12781800,  13329360,  8249850,  3031182,   612360,   52488;
  9: 8114265, 41642100, 92561130, 116405100, 90603765, 44702280, 13655628, 2361960, 177147;
%F Formula
The row polynomials satisfy (2*n-2)*P(n,x) = (5*n+10+(3*n+6)*x)*P(n-1,x) + (7+10*x+3*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[((5*n + 10 + (3*n + 6)*x)*P[n - 1, x] + (7 + 10*x + 3*x^2)*D[P[n - 1, x], x])/(2*n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = ((5*n + 10 + (3*n + 6)*x)*v[i - 1] + (7 + 10*x + 3*x^2)*deriv(v[i - 1], x))/(2*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)

polytool c48629374597

Properties

Proof chains ? JSON

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023