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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).

%I A343446 #18 Jul 23 2024 10:50:11
%S A343446 -1,1,4,40,648,14560,418880,14696640,608608000,29056867840,
%T A343446 1571364748800,94937979136000,6337884013260800,463301182536192000,
%U A343446 36806315255277568000,3157533815406530560000,290912372128665391104000,28648563542097847828480000
%N 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).
%C A343446 Based on formulas for series solutions of trinomials given in Eagle article.
%C A343446 S(p, q) = Sum_{n>=0} (a(n)*q^n)/((3^n)*(p^(4n/3))*n!)
%C A343446 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!).
%H A343446 Albert Eagle, <a href="https://www.jstor.org/stable/2303036">Series for all the roots of a trinomial equation</a>, Am. Math. Monthly, vol. 46, no. 7 (Aug. - Sep., 1939), pp. 422 - 425.
%F A343446 a(n) = 3^(n - 1)*((n + 2)/3)_(n - 1), where (x)_k is the Pochhammer symbol for Gamma(x + k)/Gamma(x).
%F A343446 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
%p A343446 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:
%p A343446 seq(a(n), n = 0..20); # _Peter Bala_, Jul 23 2024
%t A343446 Clear[a]; a=Table[3^(n - 1) Pochhammer[(n + 2)/3, n - 1], {n, 0, 20}]
%t A343446 (* 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: *)
%t A343446 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 *)
%t A343446 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}]
%Y A343446 A343445 relates similarly to the largest real root of x^3 - p*x + q.
%Y A343446 A206300 relates similarly to the largest real root of x^3 - 3*u*x + 4*u, u >= 4.
%K A343446 sign,easy
%O A343446 0,3
%A A343446 _Dixon J. Jones_, May 26 2021