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A191509 E.g.f. exp(x*sqrt(1+sin(x)^2)). 2

%I

%S 1,1,1,4,13,-4,-59,848,1625,-57968,-82679,5307072,3378277,-761466432,

%T -178851763,155538255616,13323839409,-43026868334336,-1145167641071,

%U 15502018794828800,110592144624061,-7038075176027079680,-12523284027203883,3925127762389637074944,1643266949074714633,-2635567108489125092225024

%N E.g.f. exp(x*sqrt(1+sin(x)^2)).

%H Vincenzo Librandi, <a href="/A191509/b191509.txt">Table of n, a(n) for n = 0..200</a>

%F a(n)=2*sum(j=1..(n-1)/2, binomial(n,n-2*j)*sum(k=0..j, 4^(j-k)*binomial((n-2*j)/2,k)*sum(i=0..k-1, (i-k)^(2*j)*binomial(2*k,i)*(-1)^(j+k-i))))+1.

%F If n is odd, then a(n) ~ -sin(Pi*n/2) * 2^(5/4) * log(1+sqrt(2))^(3/2-n) * n^(n-1) / exp(n). If n is even, then limit n->infinity (|a(n)| / (n! * exp(w*cosh(w)) / w^n))^(1/n) = 1, where w = 2*LambertW(sqrt(n/2)). - _Vaclav Kotesovec_, Aug 05 2014

%t CoefficientList[Series[E^(x*Sqrt[1+Sin[x]^2]), {x, 0, 20}], x] * Range[0, 20]! (* _Vaclav Kotesovec_, Aug 04 2014 *)

%o (Maxima)

%o a(n):=2*sum(binomial(n,n-2*j)*sum(4^(j-k)*binomial((n-2*j)/2,k)*sum((i-k)^(2*j)*binomial(2*k,i)*(-1)^(j+k-i),i,0,k-1),k,0,j),j,1,(n-1)/2)+1;

%Y Cf. A003727.

%K sign

%O 0,4

%A _Vladimir Kruchinin_, Jun 04 2011

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Last modified January 29 04:57 EST 2020. Contains 331335 sequences. (Running on oeis4.)