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A209299
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E.g.f.: 1 / Product_{n>=1} (cos(x^n/n) - sin(x^n/n)).
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2
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1, 1, 4, 16, 98, 650, 5492, 50468, 543252, 6375668, 83752144, 1191943168, 18563252968, 310499073352, 5598292885200, 107674197010960, 2208771882047120, 48025183073776016, 1105381958987588672, 26817991185065949440, 684717365565811694880, 18341702444087583851936
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OFFSET
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0,3
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COMMENTS
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Compare to: 1/Product_{n>=1} (cosh(x^n/n) - sinh(x^n/n)) = 1/(1-x).
Limit (a(n)/n!)^(1/n) = 4/Pi; the radius of convergence of the e.g.f. is Pi/4.
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LINKS
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Vaclav Kotesovec, Table of n, a(n) for n = 0..300
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FORMULA
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a(n) ~ c * 2^(2*n+3/2) * n! / Pi^(n+1), where c = 1 / product_{n>=2} (cos((Pi/4)^n/n) - sin((Pi/4)^n/n)) = 2.516454534521990223577410114610797032290984895329... . - Vaclav Kotesovec, Nov 04 2014
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EXAMPLE
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E.g.f.: A(x) = 1 + x + 4*x^2/2! + 16*x^3/3! + 98*x^4/4! + 650*x^5/5! +...
where A(x) = 1/((cos(x)-sin(x)) * (cos(x^2/2)-sin(x^2/2)) * (cos(x^3/3)-sin(x^3/3)) * (cos(x^4/4)-sin(x^4/4)) * (cos(x^5/5)-sin(x^5/5)) *...).
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MATHEMATICA
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With[{nmax = 50}, CoefficientList[Series[1/Product[(Cos[x^n/n] - Sin[x^n/n]), {n, 1, 200}], {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Jan 03 2018 *)
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PROG
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(PARI) {a(n)=n!*polcoeff(1/prod(k=1, n, cos(x^k/k +x*O(x^n))-sin(x^k/k +x*O(x^n))), n)}
for(n=0, 30, print1(a(n), ", "))
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CROSSREFS
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Cf. A209298.
Sequence in context: A130683 A111976 A236772 * A091040 A029985 A114023
Adjacent sequences: A209296 A209297 A209298 * A209300 A209301 A209302
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KEYWORD
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nonn
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AUTHOR
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Paul D. Hanna, Jan 17 2013
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STATUS
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approved
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