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A343446 Coefficients of the series S(p, q) for which -(p^(1/3))*S converges to the largest real root of x^4 - p*x + q, where 0 < p and 0 < q < 3*(p/4)^(4/3). 1
-1, 1, 4, 40, 648, 14560, 418880, 14696640, 608608000, 29056867840, 1571364748800, 94937979136000, 6337884013260800, 463301182536192000, 36806315255277568000, 3157533815406530560000, 290912372128665391104000, 28648563542097847828480000 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=0..17.

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(k).

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

A343445 relates similarly to the largest real root of x^3 - p*x + q.

A206300 relates similarly to the largest real root of x^3 - 3*u*x + 4*u, u >= 4.

Sequence in context: A090359 A049308 A227055 * A205671 A234294 A181088

Adjacent sequences:  A343443 A343444 A343445 * A343447 A343448 A343449

KEYWORD

sign

AUTHOR

Dixon J. Jones, May 26 2021

STATUS

approved

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Last modified May 20 11:16 EDT 2022. Contains 353871 sequences. (Running on oeis4.)