Real-rooted atlas

A103327 → basis A013610 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(-1 + 2\,n\right)\,P_{n}(x) = \left(-19 - 15\,x + 26\,n + 18\,n x\right)\,P_{n-1}(x) + \left(-16 - 28\,x - 12\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
17310
261549124
3547753297271624
449219228588614048121520
54428710518393078369906075243285856
639858111459221298511726732203715247867293798976
735872271210689316731063121034254839345102570396957218750492800

All 15 rows

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(P_{n}(x) = \left(-9 - 18\,x - 9\,x^{2}\right)\,P_{n-2}(x) + \left(10 + 6\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 7 + 3\,x\)
Technical details
Node id
n-39349a1d5687595d · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-1624ef108463d2e9)
Fingerprint
f6-b397a92ed8bf2e53

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:         7,          3;
  2:        61,         54,          9;
  3:       547,        753,        297,         27;
  4:      4921,       9228,       5886,       1404,         81;
  5:     44287,     105183,      93078,      36990,       6075,       243;
  6:    398581,    1145922,    1298511,     726732,     203715,     24786,       729;
  7:   3587227,   12106893,   16731063,   12103425,    4839345,   1025703,     96957,     2187;
  8:  32285041,  125104536,  204051564,  181193544,   94412790,  28862568,   4837644,   367416,    6561;
  9: 290565367, 1271223483, 2390354172, 2517192828, 1619293842, 649832058, 158662476, 21703788, 1358127, 19683;
%F Formula
The row polynomials satisfy (2*n-1)*P(n,x) = (26*n-19+(18*n-15)*x)*P(n-1,x) - (16+28*x+12*x^2)*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[((26*n - 19 + (18*n - 15)*x)*P[n - 1, x] - (16 + 28*x + 12*x^2)*D[P[n - 1, x], x])/(2*n - 1)]];
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] = ((26*n - 19 + (18*n - 15)*x)*v[i - 1] - (16 + 28*x + 12*x^2)*deriv(v[i - 1], x))/(2*n - 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)

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. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023