A012023 Expansion of e.g.f. cos(sin(arctan(x))) (even powers).

1, -1, 13, -421, 25369, -2449801, 346065061, -67243537453, 17192488230961, -5593309059948049, 2255588021494237501, -1103994926592923677621, 644587811150505183179593, -442516027690815793746696601

Offset

0,3

Links

Table of n, a(n) for n=0..13.

Formula

a(n) = (2*n)!*(-1)^n*sum(j=0..n, binomial(n-1,n-j)/(2*j)!). - Vladimir Kruchinin, May 19 2011

a(n) = -(12*n^2-24*n+13)*a(n-1) - 12*(n-2)*(n-1)*(2*n-3)^2*a(n-2) - 16*(n-3)*(n-2)^2*(n-1)*(2*n-5)*(2*n-3)*a(n-3). - Vaclav Kotesovec, Nov 08 2013

a(n) ~ (-1)^n * (2*n)^(2*n-1/3) * exp(3/2*(2*n)^(1/3) - 2*n) / sqrt(3) * (1 - 19/72*2^(2/3)/n^(1/3) + 553/5184*2^(1/3)/n^(2/3)). - Vaclav Kotesovec, Nov 08 2013

Example

cos(sin(arctan(x))) = 1 - (1/2!)*x^2 + (13/4!)*x^4 - (421/6!)*x^6 + (25369/8!)*x^8 - ...

Mathematica

With[{nn=30}, Take[CoefficientList[Series[Cos[Sin[ArcTan[x]]], {x, 0, nn}], x] Range[0, nn]!, {1, -1, 2}]] (* Harvey P. Dale, May 07 2012 *)

Prog

(Maxima)
a(n):=(2*n)!*(-1)^n*sum(binomial(n-1, n-j)/(2*j)!, j, 0, n); /* Vladimir Kruchinin, May 19 2011 */

Crossrefs

Sequence in context: A087872 A308341 A098890 * A081442 A100872 A012045

Adjacent sequences: A012020 A012021 A012022 * A012024 A012025 A012026

Keyword

sign

Author

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

Status

approved