OFFSET
0,3
COMMENTS
Based on formulas for series solutions of trinomials given in Eagle article.
S(p, q) = Sum_{n>=0} (a(n)*q^n)/((3^n)*(p^(4n/3))*n!)
In general, given m > 1, p > 0 and 0 < q < m*(p/(m + 1))^((m + 1)/m), the series S(m, p, q) for which (-p^(1/m))*S converges to the largest real root of x^(m + 1) - p*x + q has coefficients c(n) = m^(n - 1)*((n + m - 1)/m)_(n - 1), where (x)_k is the Pochhammer symbol for Gamma(x + k)/Gamma(k), and S(m, p, q) = Sum_{n>=0}(c(n)*q^n)/((m^n)*(p^(n*(m + 1)/m)*n!).
LINKS
Albert Eagle, Series for all the roots of a trinomial equation, Am. Math. Monthly, vol. 46, no. 7 (Aug. - Sep., 1939), pp. 422 - 425.
FORMULA
a(n) = 3^(n - 1)*((n + 2)/3)_(n - 1), where (x)_k is the Pochhammer symbol for Gamma(x + k)/Gamma(x).
a(n) = 4*(4*n - 7)*(4*n - 10)*(4*n - 13)*a(n-3) with a(0) = -1, a(1) = 1 and a(2) = 4. - Peter Bala, Jul 23 2024
MAPLE
a := proc(n) option remember; if n < 3 then [-1, 1, 4][n+1] else 4*(4*n - 7)*(4*n - 10)*(4*n - 13)*a(n-3) fi; end:
seq(a(n), n = 0..20); # Peter Bala, Jul 23 2024
MATHEMATICA
Clear[a]; a=Table[3^(n - 1) Pochhammer[(n + 2)/3, n - 1], {n, 0, 20}]
(* In general, for the series S(m, p, q) for which (-p^(1/m))*S converges to the largest real root of x^(m + 1) - p*x + q, the first n + 1 coefficients are: *)
Clear[c]; c[m_, n_] := Table[m^(k - 1) Pochhammer[(k + m - 1)/m, k - 1], {k, 0, n}](* and S(m, p, q) to n + 1 terms is given by *)
Clear[s]; s[m_, p_, q_, n_]:= Sum[c[m, n][[k + 1]]*q^k/((m^k)*(p^(k (m + 1)/m))*k!), {k, 0, n}]
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Dixon J. Jones, May 26 2021
STATUS
approved