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).
-1, 1, 3, 24, 315, 5760, 135135, 3870720, 130945815, 5109350400, 225881530875, 11158821273600, 609202488769875, 36422392637030400, 2366751668870964375, 166086110424858624000, 12517749576658530579375, 1008474862499741564928000, 86485131825133787772901875
Offset: 0
Links
- Albert Eagle, Series for all the roots of a trinomial equation, Am. Math. Monthly, 46, no. 7 (Aug. - Sep., 1939), pp. 422-425.
Programs
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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
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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}]
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
a(n) ~ 3^(3*n/2-1) * n^(n-1) / exp(n). - Amiram Eldar, Sep 02 2025
Comments