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%I #12 Jun 18 2015 16:35:42
%S 1,120,3024000,858574080000,1226178516326400000,
%T 5912338932461445120000000,75732595735526211882516480000000,
%U 2195068320271703663798288449536000000000,128322069958974226301129597680106864640000000000
%N E.g.f. satisfies: A(x) = Integral 1 + A(x)^5 dx.
%H Vaclav Kotesovec, <a href="/A258925/b258925.txt">Table of n, a(n) for n = 0..90</a>
%F E.g.f. A(x) satisfies:
%F (1) A(x) = Series_Reversion( Integral 1/(1+x^5) dx ).
%F Let C(x) = S'(x) such that S(x) = Series_Reversion( Integral 1/(1-x^5)^(1/5) dx ) is the e.g.f. of A258924, then e.g.f. A(x) of this sequence satisfies:
%F (2) A(x) = S(x)/C(x),
%F (3) A(x) = Integral 1/C(x)^5 dx.
%F From _Vaclav Kotesovec_, Jun 18 2015: (Start)
%F a(n) ~ 5^(5*n+5/2) * ((5-sqrt(5))/8)^(5*n/2+5/8) * (5*n)! * n^(1/4) / (sqrt(2) * Gamma(1/4) * Pi^(5*n+5/4)).
%F a(n) ~ 5^(5*n+5/2) * sin(Pi/5)^(5*n+5/4) * (5*n)! * n^(1/4) / (sqrt(2) * Gamma(1/4) * Pi^(5*n+5/4)).
%F (End)
%e E.g.f.: A(x) = x + 120*x^6/6! + 3024000*x^11/11! + 858574080000*x^16/16! +...
%e where Series_Reversion(A(x)) = x - x^6/6 + x^11/11 - x^16/16 + x^21/21 +...
%e Also, A(x) = S(x)/C(x) where
%e S(x) = x - 24*x^6/6! - 169344*x^11/11! - 25255286784*x^16/16! - 23089632627769344*x^21/21! +...+ A258924(n)*x^(5*n+1)/(5*n+1)! +...
%e C(x) = 1 - 24*x^5/5! - 169344*x^10/10! - 25255286784*x^15/15! - 23089632627769344*x^20/20! +...+ A258924(n)*x^(5*n)/(5*n)! +...
%e such that C(x)^5 + S(x)^5 = 1.
%o (PARI) /* E.g.f. Series_Reversion( Integral 1/(1+x^5) dx ): */
%o {a(n) = local(A=x); A = serreverse( intformal( 1/(1 + x^5 + O(x^(5*n+2))) ) ); (5*n+1)!*polcoeff(A, 5*n+1)}
%o for(n=0, 20, print1(a(n), ", "))
%o (PARI) /* E.g.f. A(x) = Integral 1 + A(x)^5 dx.: */
%o {a(n) = local(A=x); for(i=1, n+1, A = intformal( 1 + A^5 + O(x^(5*n+2)) )); (5*n+1)!*polcoeff(A, 5*n+1)}
%o for(n=0, 20, print1(a(n), ", "))
%Y Cf. A258924, A258880, A258901, A258927, A259112, A259113, A258971.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jun 15 2015
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