OFFSET
0,3
COMMENTS
Based on formulas for series solutions of trinomials given in Eagle article.
LINKS
Albert Eagle, Series for all the roots of a trinomial equation, Am. Math. Monthly, 46, no. 7 (Aug. - Sep., 1939), pp. 422-425.
FORMULA
a(n) = 2^(n - 1) * Gamma((3*n - 1)/2) / Gamma((n + 1)/2).
a(n) = 2^(n - 1) * ((n + 1)/2)_(n - 1), where (x)_k is the Pochhammer symbol for Gamma(x + k) / Gamma(k).
a(n) = 3*A113551(n-1) for n>=2. - Hugo Pfoertner, Apr 16 2021
E.g.f.: (sqrt(3)*sin(arcsin(3*sqrt(3)*x)/3) - 3*cos(arcsin(3*sqrt(3)*x)/3))/3. - Stefano Spezia, May 23 2021
a(n) = 3*(3*n - 5)*(3*n - 7)*a(n-2) with a(0) = -1, a(1) = 1 and a(2) = 3. - Peter Bala, Jul 23 2024
MAPLE
a := proc(n) option remember; if n = 1 then 1 elif n = 2 then 3 else 3*(3*n - 5)*(3*n - 7)*a(n-2) fi; end:
seq(a(n), n = 1..20); # Peter Bala, Jul 23 2024
MATHEMATICA
Clear[a]; a = Table[2^(n - 1)Gamma[(3*n - 1)/2]/Gamma[(n + 1)/2], {n, 0, 20}] (* or equivalently *)
Clear[a]; a = Table[2^(n - 1)Pochhammer[(n + 1)/2, n - 1], {n, 0, 20}]
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Dixon J. Jones, Apr 15 2021
STATUS
approved