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A012027 E.g.f. cosh(sin(arctan(x))) = cosh(x/sqrt(1+x^2)) (even powers only). 0
1, 1, -11, 301, -15287, 1239481, -146243459, 23567903269, -4951201340399, 1307274054385393, -420773143716828539, 160635990248839962781, -70764171306270411101351, 34822234810202848704345001 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

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

FORMULA

a(n) = ((2*n)!*sum(k=0..n, C(n-1,n-k)/(2*k)!*(-1)^(n-k))). - Vladimir Kruchinin, Jun 17 2011

E.g.f.: cosh(x/sqrt(1+x^2)) = 1 + x^2/(G(0) - x^2) where G(k)= (2*k+2)*(2*k+1)*(1+x^2) + x^2 - (2*k+2)*(2*k+1)*x^2*(1+x^2)/G(k+1); (continued fraction, Euler's kind, 1-step). - Sergei N. Gladkovskii, Aug 06 2012

D-finite with recurrence: a(n) = -(12*n^2 - 24*n + 11)*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 09 2013

Lim sup n->infinity |a(n)|/(2^(2*n+2/3) * exp(3/4*(2*n)^(1/3)-2*n) * n^(2*n-1/3) / sqrt(3)) = 1. - Vaclav Kotesovec, Nov 09 2013

EXAMPLE

cosh(sin(arctan(x))) = 1+1/2!*x^2-11/4!*x^4+301/6!*x^6-15287/8!*x^8...

MATHEMATICA

Table[n!*SeriesCoefficient[Cosh[x/Sqrt[1+x^2]], {x, 0, n}], {n, 0, 40, 2}] (* Vaclav Kotesovec, Nov 08 2013 *)

PROG

(Maxima)

a(n):=((2*n)!*sum(binomial(n-1, n-k)/(2*k)!*(-1)^(n-k), k, 0, n)); [Vladimir Kruchinin, Jun 17 2011]

CROSSREFS

Sequence in context: A001538 A101269 A012184 * A279181 A002114 A012192

Adjacent sequences: A012024 A012025 A012026 * A012028 A012029 A012030

KEYWORD

sign

AUTHOR

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

STATUS

approved

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Last modified February 3 01:46 EST 2023. Contains 360024 sequences. (Running on oeis4.)