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A343445 Coefficients of the series S(p, q) for which (-sqrt(p))*S converges to the largest real root of x^3 - p*x + q for 0 < p and 0 < q < 2*(p/3)^(3/2).

%I A343445 #35 Sep 02 2025 04:06:52
%S A343445 -1,1,3,24,315,5760,135135,3870720,130945815,5109350400,225881530875,
%T A343445 11158821273600,609202488769875,36422392637030400,2366751668870964375,
%U A343445 166086110424858624000,12517749576658530579375,1008474862499741564928000,86485131825133787772901875
%N A343445 Coefficients of the series S(p, q) for which (-sqrt(p))*S converges to the largest real root of x^3 - p*x + q for 0 < p and 0 < q < 2*(p/3)^(3/2).
%C A343445 Based on formulas for series solutions of trinomials given in Eagle article.
%H A343445 Albert Eagle, <a href="https://www.jstor.org/stable/2303036">Series for all the roots of a trinomial equation</a>, Am. Math. Monthly, 46, no. 7 (Aug. - Sep., 1939), pp. 422-425.
%F A343445 a(n) = 2^(n - 1) * Gamma((3*n - 1)/2) / Gamma((n + 1)/2).
%F A343445 a(n) = 2^(n - 1) * ((n + 1)/2)_(n - 1), where (x)_k is the Pochhammer symbol for Gamma(x + k) / Gamma(k).
%F A343445 a(n) = 3*A113551(n-1) for n>=2. - _Hugo Pfoertner_, Apr 16 2021
%F A343445 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
%F A343445 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
%F A343445 a(n) ~ 3^(3*n/2-1) * n^(n-1) / exp(n). - _Amiram Eldar_, Sep 02 2025
%p A343445 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:
%p A343445 seq(a(n), n = 1..20); # _Peter Bala_, Jul 23 2024
%t A343445 Clear[a]; a = Table[2^(n - 1)Gamma[(3*n - 1)/2]/Gamma[(n + 1)/2], {n, 0, 20}] (* or equivalently *)
%t A343445 Clear[a]; a = Table[2^(n - 1)Pochhammer[(n + 1)/2, n - 1], {n, 0, 20}]
%Y A343445 Cf. A113551, A343446.
%K A343445 sign,easy,changed
%O A343445 0,3
%A A343445 _Dixon J. Jones_, Apr 15 2021