Real-rooted atlas

A116414 → basis A007318 → basis A007318

nonnegative coefficients

Proved for all n all roots real, in (−10, −2)coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(2\,n\right)\,P_{n}(x) = \left(6 + x + 6\,n + n x\right)\,P_{n-1}(x) + \left(24 + 12\,x + x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1617
23312146
3180102181301
49817562072411969
553465211194434830112880
6291333434416254396052536184253
7158760219564126036389707020738421551131

All 15 rows

Other recurrences

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

\(P_{n}(x) = \left(-3\right)\,P_{n-2}(x) + \left(6 + x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 6 + x\)
Technical details
Node id
n-c77c01e985dfc77d · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-e9be29fb091d3543)
Fingerprint
f6-a6ac24daa8213573

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:       6,       1;
  2:      33,      12,       1;
  3:     180,     102,      18,       1;
  4:     981,     756,     207,      24,      1;
  5:    5346,    5211,    1944,     348,     30,      1;
  6:   29133,   34344,   16254,    3960,    525,     36,     1;
  7:  158760,  219564,  126036,   38970,   7020,    738,    42,    1;
  8:  865161, 1373112,  927018,  347976,  79515,  11340,   987,   48,  1;
  9: 4714686, 8445141, 6557112, 2897964, 804006, 145341, 17136, 1272, 54, 1;
%F Formula
The row polynomials satisfy 2*n*P(n,x) = (6*n+6+(n+1)*x)*P(n-1,x) + (24+12*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[((6*n + 6 + (n + 1)*x)*P[n - 1, x] + (24 + 12*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] = ((6*n + 6 + (n + 1)*x)*v[i - 1] + (24 + 12*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