A012091 Even coefficients in expansion of e.g.f. cos(arcsin(arctan(x))).

1, -1, 5, -109, 4521, -330681, 36468717, -5721101541, 1205289090513, -328897528901361, 112769846051251797, -47472161654530776285, 24068680135210975714425, -14466929567049590650593705, 10171963709756690616315847485, -8271551427684042569818592399445, 7702642043364121393979444828587425

Offset

0,3

Links

Table of n, a(n) for n=0..16.

Peter Bala, The method of Graves for computing inverse functions.

Formula

a(n) = (2*n)!*sum(k=1..2*n, ((2*k)!*binomial(2*k-2,k-1)*(-1)^(n+k+1)*sum(i=2*k..2*n, (2^(i+1)*Stirling1(i,2*k)*binomial(2*n-1,i-1))/i!))/(k*2^(4*k))) with n>0, a(0)=1. - Vladimir Kruchinin, Oct 08 2012

E.g.f.: A(x) = sqrt(1 - arctan(x)^2). 1/A(x) is the e.g.f. of A012097. - Peter Bala, Jan 28 2026

a(n) ~ (-1)^n * (2*n-1)! * (1 - 2/log(n)^2 + (4*gamma + 8*log(2))/log(n)^3 + (Pi^2 - 6*gamma^2 - 24*gamma*log(2) - 24*log(2)^2 + 6)/log(n)^4), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jan 29 2026

Example

cos(arcsin(arctan(x))) = 1 - 1/2!*x^2 + 5/4!*x^4 - 109/6!*x^6 + 4521/8!*x^8 ...

Maple

d := proc (n, x) option remember; if n = 0 then sqrt(1 - x^2) else simplify( (cos(x))^2 * (diff(d(n-1, x), x))) end if end proc:
seq(eval(d(2*n, x), x = 0), n = 0..20); # Peter Bala, Jan 28 2026

Mathematica

With[{nn=30}, Take[CoefficientList[Series[Cos[ArcSin[ArcTan[x]]], {x, 0, nn}], x] Range[0, nn]!, {1, -1, 2}]] (* Harvey P. Dale, Feb 07 2015 *)
nmax = 20; CoefficientList[Series[Sqrt[1 - ArcTan[Sqrt[x]]^2], {x, 0, nmax}], x] * (2*Range[0, nmax])! (* Vaclav Kotesovec, Jan 29 2026 *)

Prog

(Maxima) a[n]:=if n=0 then 1 else (2*n)!*sum(((2*k)!*binomial(2*k-2, k-1)*(-1)^(n+k+1)*sum((2^(i+1)*stirling1(i, 2*k)*binomial(2*n-1, i-1))/i!, i, 2*k, 2*n))/(k*2^(4*k)), k, 1, 2*n); makelist(a[n], n, 0, 13); /* Vladimir Kruchinin, Oct 08 2012 */

Crossrefs

Cf. A012254, A012097.

Sequence in context: A376462 A014180 A012122 * A322896 A296743 A358781

Adjacent sequences: A012088 A012089 A012090 * A012092 A012093 A012094

Keyword

sign,easy

Author

Patrick Demichel

Extensions

Terms a(14) - a(16) added by Peter Bala, Jan 31 2026

Status

approved