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%I #15 Feb 15 2019 10:04:03
%S 1,1,5,85,2825,151625,12098125,1339476125,196410020625,37062144900625,
%T 8772471210303125,2519410212081953125,854580849916226265625,
%U 348321081954065679265625,175848355607462110288828125,95409938858134878259950078125,39148051664861004366855145390625,29871598369545793078681814025390625,108819532565544659211041992639176953125
%N E.g.f. A(x) satisfies: A(A(x)) = sinh( arcsin(x) ).
%C a(n) modulo 8 has period 4: [1, 1, 5, 5].
%C a(n) = 5 (mod 10) for n > 2.
%C First negative term is a(21).
%H Paul D. Hanna, <a href="/A318635/b318635.txt">Table of n, a(n) for n = 1..100</a>
%e E.g.f.: A(x) = x + x^3/3! + 5*x^5/5! + 85*x^7/7! + 2825*x^9/9! + 151625*x^11/11! + 12098125*x^13/13! + 1339476125*x^15/15! + 196410020625*x^17/17! + 37062144900625*x^19/19! + 8772471210303125*x^21/21! + 2519410212081953125*x^23/23! + 854580849916226265625*x^25/25! +...
%e such that
%e A(A(x)) = x + 2*x^3 + 20*x^5 + 520*x^7 + 26000*x^9 + 2132000*x^11 + 260104000*x^13 + 44217680000*x^15 + ... + [Product_{k=1..n-1} ((2*k-1)^2 + 1)] * x^(2*n-1)/(2*n-1)! + ...
%e which equals sinh( arcsin(x) ).
%e RELATED SERIES.
%e Given e.g.f. F(x) of A280790 such that sin( F( sinh( F(x) ) ) ) = x,
%e then A(x) = sinh( F(x) ) where
%e F(x) = x + 4*x^5/5! + 2320*x^9/9! + 9857600*x^13/13! + 159122080000*x^17/17! + 7098806416000000*x^21/21! + 686863244097538560000*x^25/25! + ... + A280790(n)*x^(4*n-3)/(4*n-3)! + ...
%e also sin( F( A(x) ) ) = x.
%o (PARI) {a(n) = my(A=[1]); for(i=1,n, A = concat(A,[0,0]); A[#A] = polcoeff(sinh(asin(x +x*O(x^#A))) - subst(x*Ser(A),x,x*Ser(A)),#A)/2 ); (2*n-1)!*A[2*n-1]}
%o for(n=1,20,print1(a(n),", "))
%Y Cf. A280790, A101927.
%K sign
%O 1,3
%A _Paul D. Hanna_, Sep 05 2018
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