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%I #8 Jan 14 2017 14:20:37
%S 1,-1,-1,-13,-367,-18549,-1465953,-166262441,-25526158559,
%T -5088504559209,-1277203721141441,-394351281523218693,
%U -147069662568684159055,-65255038602423680990301,-33992871584988519888865825,-20539580068386370855911967393,-14240009948449682448965044873663,-11224227759618581623496389602591953,-9984932631658989030110444663072663937,-9963529315662216629464111409263738683133
%N E.g.f. sqrt(1 - F(x)^2), where F(F(x)) = sin(x).
%C This sequence appears to consist entirely of integers.
%H Paul D. Hanna, <a href="/A280796/b280796.txt">Table of n, a(n) for n = 0..100</a>
%F E.g.f. A(x) satisfies: A(F(x)) = cos(x), where F(F(x)) = sin(x).
%e E.g.f.: A(x) = 1 - x^2/2! - x^4/4! - 13*x^6/6! - 367*x^8/8! - 18549*x^10/10! - 1465953*x^12/12! - 166262441*x^14/14! - 25526158559*x^16/16! - 5088504559209*x^18/18! - 1277203721141441*x^20/20! - 394351281523218693*x^22/22! - 147069662568684159055*x^24/24! - 65255038602423680990301*x^26/26! +...
%e such that A(x) = sqrt(1 - F(x)^2) where F(F(x)) = sin(x) and F(x) begins:
%e F(x) = x - 1/2*x^3/3! - 3/4*x^5/5! - 53/8*x^7/7! - 1863/16*x^9/9! - 92713/32*x^11/11! - 3710155/64*x^13/13! + 594673187/128*x^15/15! + 329366540401/256*x^17/17! + 104491760828591/512*x^19/19! + 19610322215706989/1024*x^21/21! - 5244397496803513989/2048*x^23/23! - 7592640928150019948759/4096*x^25/25! +...
%e Also, e.g.f. A(x) satisfies: A(F(x)) = cos(x).
%o (PARI) {a(n)=local(A, B, F); F=sin(x+O(x^(2*n+1))); A=F; for(i=0, 2*n-1, B=serreverse(A); A=(A+subst(B, x, F))/2); (2*n)!*polcoeff(sqrt(1-A^2), 2*n, x)}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A098932, A280795.
%K sign
%O 0,4
%A _Paul D. Hanna_, Jan 13 2017
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