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

1, 1, 1, 4, 13, -4, -59, 848, 1625, -57968, -82679, 5307072, 3378277, -761466432, -178851763, 155538255616, 13323839409, -43026868334336, -1145167641071, 15502018794828800, 110592144624061, -7038075176027079680, -12523284027203883, 3925127762389637074944, 1643266949074714633, -2635567108489125092225024

Offset: 0

Author

Vladimir Kruchinin, Jun 04 2011

Keywords

• sign

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A003727.

Programs

Mathematica

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

Maxima

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;

Formula

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.

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