Real-rooted atlas

A110441 → basis A007318 → basis A007318

nonnegative coefficients

Proved for all n all roots real, in (−8, −2)coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(2\,n\right)\,P_{n}(x) = \left(5 + x + 5\,n + n x\right)\,P_{n-1}(x) + \left(17 + 10\,x + x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1516
22310134
310571151192
44794401442011084
52185253711302422516120
69967139907899230036530134552
7454657484351225189154075513351195072

All 15 rows

Other recurrences

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

\(P_{n}(x) = \left(-2\right)\,P_{n-2}(x) + \left(5 + x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 5 + x\)
Technical details
Node id
n-48886cd788ae2641 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-aced1b7f4f5c7054)
Fingerprint
f6-3cb9d9bca2345232

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:      5,       1;
  2:     23,      10,       1;
  3:    105,      71,      15,      1;
  4:    479,     440,     144,     20,      1;
  5:   2185,    2537,    1130,    242,     25,     1;
  6:   9967,   13990,    7899,   2300,    365,    30,    1;
  7:  45465,   74843,   51225,  18915,   4075,   513,   35,   1;
  8: 207391,  391700,  315170, 141200,  38560,  6580,  686,  40,  1;
  9: 946025, 2016205, 1865100, 983340, 325850, 70434, 9940, 884, 45, 1;
%F Formula
The row polynomials satisfy 2*n*P(n,x) = (5*n+5+(n+1)*x)*P(n-1,x) + (17+10*x+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[((5*n + 5 + (n + 1)*x)*P[n - 1, x] + (17 + 10*x + x^2)*D[P[n - 1, x], x])/(2*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] = ((5*n + 5 + (n + 1)*x)*v[i - 1] + (17 + 10*x + x^2)*deriv(v[i - 1], x))/(2*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

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