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A230114
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E.g.f. 1/(1 - sin(8*x))^(1/8).
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4
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1, 1, 9, 89, 1521, 32401, 869049, 27608489, 1019581281, 42824944801, 2017329504489, 105299243488889, 6032850630082641, 376363074361201201, 25396689469918450329, 1843101478742259481289, 143145930384321475601601, 11846611289341729822881601, 1040750126963789832859930569
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OFFSET
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0,3
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COMMENTS
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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
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LINKS
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FORMULA
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E.g.f. A(x) satisfies:
(1) A(x) = (cos(4*x) - sin(4*x))^(-1/4).
(2) A(x)^4/A(-x)^4 = 1/cos(8*x) + tan(8*x).
(3) A(x) = exp( Integral A(x)^4/A(-x)^4 dx ).
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].
a(n) ~ n! * 2^(4*n+3/8) / (Gamma(1/4) * n^(3/4) * Pi^(n+1/4)). - Vaclav Kotesovec, Jan 03 2014
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EXAMPLE
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E.g.f.: A(x) = 1 + x + 9*x^2/2! + 89*x^3/3! + 1521*x^4/4! + 32401*x^5/5! + ...
where A(x)^4 = 1 + 4*x + 48*x^2/2! + 704*x^3/3! + 14592*x^4/4! + 369664*x^5/5! + ...
and 1/A(x)^4 = 1 - 4*x - 16*x^2/2! + 64*x^3/3! + 256*x^4/4! - 1024*x^5/5! + ...
which illustrates 1/A(x)^4 = cos(4*x) - sin(4*x).
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.
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MATHEMATICA
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CoefficientList[Series[1/(1-Sin[8*x])^(1/8), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jan 03 2014 *)
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PROG
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(PARI) {a(n)=local(X=x+x*O(x^n)); n!*polcoeff((cos(4*X)-sin(4*X))^(-1/4), n)}
for(n=0, 20, print1(a(n), ", "))
(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)}
for(n=0, 20, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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