A165233 Signed denominators of terms in series expansion of cos(x)+sin(x).

1, 1, -2, -6, 24, 120, -720, -5040, 40320, 362880, -3628800, -39916800, 479001600, 6227020800, -87178291200, -1307674368000, 20922789888000, 355687428096000, -6402373705728000, -121645100408832000, 2432902008176640000

Offset: 0

Author

Jaume Oliver Lafont, Sep 09 2009

Keywords

• frac
• sign
• easy

Comments

Sum(n>=0,1/a(n))=cos(1)+sin(1).

Sum(n>=0,(Pi/4)^n/a(n))=sqrt(2).

Numerators are in A000012. - Alois P. Heinz, Jan 20 2016

Crossrefs

Cf. A000012, A000142, A133942, A057077, A143623, A002193.

Programs

Mathematica

Sign@ # Denominator@ # & /@ CoefficientList[Series[Cos@ x + Sin@ x, {x, 0, 20}], x] (* Michael De Vlieger, Oct 08 2016 *)

PARI

a(n)=(-1)^(n\2)*n!

Formula

a(n)=(-1)^floor(n/2)*n! = A057077(n)*A000142(n).

a(n)=(sin(n*Pi/2)+cos(n*Pi/2))*n!.

a(n)=sqrt(2)*sin((2n+1)*Pi/4)*n!.

a(n)=sqrt(2)*cos((2n-1)*Pi/4)*n!.

G.f. Q(0) where Q(k)= 1 + x*(4*k+1)/(1 + 2*x*(2*k+1)/(1 - 2*x*(2*k+1) - x*(4*k+3)/(1 + x*(4*k+3) - 4*x*(k+1)/(4*x*(k+1) - 1/Q(k+1))))); (continued fraction, 3rd kind, 5-step). - Sergei N. Gladkovskii, Aug 15 2012

E.g.f.: (1 + x)/(1 + x^2). - Ilya Gutkovskiy, Oct 08 2016