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A209299 E.g.f.: 1 / Product_{n>=1} (cos(x^n/n) - sin(x^n/n)). 2

%I #17 Jan 04 2018 17:31:34

%S 1,1,4,16,98,650,5492,50468,543252,6375668,83752144,1191943168,

%T 18563252968,310499073352,5598292885200,107674197010960,

%U 2208771882047120,48025183073776016,1105381958987588672,26817991185065949440,684717365565811694880,18341702444087583851936

%N E.g.f.: 1 / Product_{n>=1} (cos(x^n/n) - sin(x^n/n)).

%C Compare to: 1/Product_{n>=1} (cosh(x^n/n) - sinh(x^n/n)) = 1/(1-x).

%C Limit (a(n)/n!)^(1/n) = 4/Pi; the radius of convergence of the e.g.f. is Pi/4.

%H Vaclav Kotesovec, <a href="/A209299/b209299.txt">Table of n, a(n) for n = 0..300</a>

%F 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

%e E.g.f.: A(x) = 1 + x + 4*x^2/2! + 16*x^3/3! + 98*x^4/4! + 650*x^5/5! +...

%e 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)) *...).

%t 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 *)

%o (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)}

%o for(n=0,30,print1(a(n),", "))

%Y Cf. A209298.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jan 17 2013

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