A012290 Expansion of e.g.f. arctan(sin(x)*exp(x)).

0, 1, 2, 0, -24, -140, -8, 8960, 92736, 12240, -14154208, -210017280, -50776704, 58549241920, 1133642065792, 435228385280, -500888609593344, -11981516394489600, -6690495105515008, 7684815276420464640

Offset

0,3

Links

G. C. Greubel, Table of n, a(n) for n = 0..423

Formula

a(n) = Sum_{m=0..n/2} ((4^(-m)*Sum_{j=m..n/2} (binomial(n+1,2*j+1)*((2*m+1)^(n-2*j-1)*sum(i=0..m+1/2, (2*i-2*m-1)^(2*j+1)*(-1)^(j+1-i)*binomial(2*m+1,i)))))). - Vladimir Kruchinin, Jun 30 2011

Example

arctan(sin(x)*exp(x)) = x + (2/2!)*x^2 - (24/4!)*x^4 - (140/5!)*x^5 - (8/6!)*x^6 + ...

Mathematica

CoefficientList[Series[ArcTan[Sin[x]*Exp[x]], {x, 0, 20}], x]*Range[0, 20]! (* Vaclav Kotesovec, Jan 02 2014 *)

Prog

(Maxima)
a(n):=sum((4^(-m)*sum(binomial(n+1, 2*j+1)*((2*m+1)^(n-2*j-1)*sum((2*i-2*m-1)^(2*j+1)*(-1)^(j+1-i)*binomial(2*m+1, i), i, 0, m+1/2)), j, m, (n)/2)), m, 0, (n)/2); /* Vladimir Kruchinin, Jun 30 2011 */
(PARI) x='x+O('x^30); concat([0], Vec(serlaplace(atan(sin(x)*exp(x))))) \\ G. C. Greubel, Oct 26 2018
(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Arctan(Sin(x)*Exp(x)) )); [0] cat [Factorial(n)*b[n]: n in [1..m-1]]; // G. C. Greubel, Oct 26 2018

Crossrefs

Sequence in context: A052607 A052602 A012588 * A012591 A319221 A009799

Adjacent sequences: A012287 A012288 A012289 * A012291 A012292 A012293

Keyword

sign

Author

Patrick Demichel (patrick.demichel(AT)hp.com)

Extensions

Missing a(0)=0 prepended by Vaclav Kotesovec, Jan 02 2014

Status

approved