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A209298 E.g.f.: Product_{n>=1} (cos(x^n/n) + sin(x^n/n)). 2

%I #14 Jan 04 2018 17:31:27

%S 1,1,0,4,6,30,348,1580,12516,114884,1375776,12239280,160067160,

%T 1966619512,28104385008,428735710000,6769181533968,110402248461840,

%U 2070626881211136,38342125010072384,764180537501729376,16185744192232110560,354756690964676468160

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

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

%H Vaclav Kotesovec, <a href="/A209298/b209298.txt">Table of n, a(n) for n = 0..400</a>

%e E.g.f.: A(x) = 1 + x + 4*x^3/3! + 6*x^4/4! + 30*x^5/5! + 348*x^6/6! +...

%e where A(x) = (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)) *...

%e What is the limit a(n)/n! = ?

%e Example:

%e a(1000)/1000! = 0.2942615679517020268...

%e a(2000)/2000! = 0.2939735835938621667...

%e a(3000)/3000! = 0.2938768494981674721...

%e a(4000)/4000! = 0.2938283311328618257...

%e a(5000)/5000! = 0.2937991678075013564...

%e a(6000)/6000! = 0.2937797033327244435...

%t With[{nmax = 50}, CoefficientList[Series[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(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. A209299.

%K nonn

%O 0,4

%A _Paul D. Hanna_, Jan 17 2013

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Last modified April 14 00:20 EDT 2024. Contains 371652 sequences. (Running on oeis4.)