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A158111 E.g.f.: sm^-1(x) = Sum_{n>=0} a(n)*x^(3n+1)/(3n+1)!; a(n) = coefficient of x^(3n+1)/(3n+1)! in the Maclaurin expansion of the inverse of the Dixon elliptic function sm(x,0).

%I A158111 #22 Dec 10 2024 05:38:50
%S A158111 1,4,400,179200,216832000,552487936000,2554704216064000,
%T A158111 19415752042086400000,225960522265801523200000,
%U A158111 3818732826292045742080000000,89923520593525093134499840000000,2854532237720860556461562920960000000,118891267701073842176624095657984000000000
%N A158111 E.g.f.: sm^-1(x) = Sum_{n>=0} a(n)*x^(3n+1)/(3n+1)!; a(n) = coefficient of x^(3n+1)/(3n+1)! in the Maclaurin expansion of the inverse of the Dixon elliptic function sm(x,0).
%C A158111 sm(x) = sm(x,0) satisfies: Integral_{y=0..sm(x,0)} dy/(1-y^3)^(2/3) = x.
%F A158111 a(n) = Product_{k=1..n} (3k-2)*(3k-1)^2 for n > 0 with a(0)=1.
%F A158111 E.g.f.: Sum_{n>=0} a(n)*x^(3m)/(3m)! = 1/(1-x^3)^(2/3).
%F A158111 From _Peter Bala_, Feb 22 2015: (Start)
%F A158111 a(n) = (n - 1/3)! * (3*n)!/( (-1/3)! * n! ).
%F A158111 a(n) = Product {k = 1..3*n} (k - 0^(k mod 3)), where we apply the usual convention that 0^0 = 1. Cf. A255406. (End)
%F A158111 a(n) ~ Gamma(1/3) * 3^(3*n + 1) * n^(3*n + 1/6) / (sqrt(2*Pi) * exp(3*n)). - _Vaclav Kotesovec_, Apr 10 2018
%e A158111 E.g.f.: 1/(1-x^3)^(2/3) = 1 + 4*x^3/3! + 400*x^6/6! + 179200*x^9/9! + ...
%e A158111 E.g.f.: sm^-1(x) = x + 4*x^4/4! + 400*x^7/7! + 179200*x^10/10! + ...
%e A158111 sm(x) = x - 4*x^4/4! + 160*x^7/7! - 20800*x^10/10! + 6476800*x^13/13! + ...
%p A158111 a(n):= mul(k-0^(mod(k,3)),k=1..3*n):seq(a(n), n = 0 .. 12);
%p A158111 # _Peter Bala_, Feb 22 2015
%t A158111 Join[{1},Table[Product[(3k-2)(3k-1)^2,{k,n}],{n,14}]] (* _Harvey P. Dale_, May 19 2012 *)
%t A158111 a[k_] := Pochhammer[2/3, k] (3 k)!/k!; Array[a, 15, 0] (* _Jan Mangaldan_, Jan 06 2017 *)
%o A158111 (PARI) a(n)=prod(k=1,n,(3*k-2)*(3*k-1)^2)
%Y A158111 Cf. A104133, A004988, A255406.
%K A158111 nonn,easy
%O A158111 0,2
%A A158111 _Paul D. Hanna_, Mar 18 2009
%E A158111 More terms from _Harvey P. Dale_, May 19 2012