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%I #22 Sep 23 2022 17:06:49
%S 1,1,9,89,1521,32401,869049,27608489,1019581281,42824944801,
%T 2017329504489,105299243488889,6032850630082641,376363074361201201,
%U 25396689469918450329,1843101478742259481289,143145930384321475601601,11846611289341729822881601,1040750126963789832859930569
%N E.g.f. 1/(1 - sin(8*x))^(1/8).
%C Generally, for e.g.f. 1/(1-sin(p*x))^(1/p) is a(n) ~ n! * 2^(n+3/p) * p^n / (Gamma(2/p) * n^(1-2/p) * Pi^(n+2/p)). - _Vaclav Kotesovec_, Jan 03 2014
%F E.g.f. A(x) satisfies:
%F (1) A(x) = (cos(4*x) - sin(4*x))^(-1/4).
%F (2) A(x)^4/A(-x)^4 = 1/cos(8*x) + tan(8*x).
%F (3) A(x) = exp( Integral A(x)^4/A(-x)^4 dx ).
%F O.g.f.: 1/G(0) where G(k) = 1 - (8*k+1)*x - 8*(k+1)*(4*k+1)*x^2/G(k+1) [continued fraction formula from A144015 due to _Sergei N. Gladkovskii_].
%F a(n) ~ n! * 2^(4*n+3/8) / (Gamma(1/4) * n^(3/4) * Pi^(n+1/4)). - _Vaclav Kotesovec_, Jan 03 2014
%e E.g.f.: A(x) = 1 + x + 9*x^2/2! + 89*x^3/3! + 1521*x^4/4! + 32401*x^5/5! + ...
%e where A(x)^4 = 1 + 4*x + 48*x^2/2! + 704*x^3/3! + 14592*x^4/4! + 369664*x^5/5! + ...
%e and 1/A(x)^4 = 1 - 4*x - 16*x^2/2! + 64*x^3/3! + 256*x^4/4! - 1024*x^5/5! + ...
%e which illustrates 1/A(x)^4 = cos(4*x) - sin(4*x).
%e O.g.f.: 1/(1-x - 8*1*1*x^2/(1-9*x - 8*2*5*x^2/(1-17*x - 8*3*9*x^2/(1-25*x - 8*4*13*x^2/(1-33*x - 8*5*17*x^2/(1-...)))))), a continued fraction.
%t CoefficientList[Series[1/(1-Sin[8*x])^(1/8), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Jan 03 2014 *)
%o (PARI) {a(n)=local(X=x+x*O(x^n)); n!*polcoeff((cos(4*X)-sin(4*X))^(-1/4), n)}
%o for(n=0, 20, print1(a(n), ", "))
%o (PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=exp(intformal(A^4/subst(A^4, x, -x)))); n!*polcoeff(A, n)}
%o for(n=0, 20, print1(a(n), ", "))
%Y Cf. A001586 (p=2), A007788 (p=3), A144015 (p=4), A230134 (p=5), A227544 (p=6), A235128 (p=7).
%K nonn
%O 0,3
%A _Paul D. Hanna_, Dec 20 2013
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