A012079 Even coefficients in expansion of e.g.f. cos(arcsin(tan(x))).

1, -1, -11, -301, -16631, -1620601, -250557251, -56629836901, -17602836565871, -7193368568377201, -3735581618747946491, -2401310609311293289501, -1871136400199563216523111, -1737384477758027314448941801, -1895055889762314322374487239731, -2399005943025880293676172824022101

Offset

0,3

Links

Table of n, a(n) for n=0..15.

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

Formula

a(n) = -2*Sum_{k = 1..2*n} C(k-1)*Sum_{j = 2*k..2*n} binomial(j-1,2*k-1)*j!*2^(2*n-j-2*k)*(-1)^((n+k)+j)*Stirling2(2*n,j) with n>0, a(0)=1, C(n)=A000108(n) (Catalan numbers). - Vladimir Kruchinin, Oct 08 2012

E.g.f.: A(x) = sqrt(1 - tan(x)^2). 1/A(x) is the e.g.f. of A012085. - Peter Bala, Jan 28 2026

a(n) ~ -2^(6*n - 1/2) * n^(2*n-1) / (Pi^(2*n - 1/2) * exp(2*n)). - Vaclav Kotesovec, Jan 29 2026

Example

cos(arcsin(tan(x))) = 1 - 1/2!*x^2 - 11/4!*x^4 - 301/6!*x^6 - 16631/8!*x^8 ...

Maple

d := proc (n, x) option remember; if n = 0 then sqrt(1 - x^2) else simplify((1 + x^2) *(diff(d(n-1, x), x))) end if end proc:
seq(eval(d(2*n, x), x = 0), n = 0..20);

Mathematica

With[{nn=30}, Take[CoefficientList[Series[Cos[ArcSin[Tan[x]]], {x, 0, nn}], x] Range[0, nn]!, {1, -1, 2}]] (* Harvey P. Dale, Jan 23 2019 *)
nmax = 20; CoefficientList[Series[Sqrt[1 - Tan[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*sum(binomial(2*k-2, k-1)/k*sum(binomial(j-1, 2*k-1)*j!*2^(2*n-j+(-2)*k)*(-1)^(n+k+j)*stirling2(2*n, j), j, 2*k, 2*n), k, 1, 2*n); makelist(a[n], n, 0, 12); /* Vladimir Kruchinin, Oct 08 2012 */

Crossrefs

Bisection of A012253.

Cf. A012085, A012097.

Sequence in context: A279181 A002114 A012192 * A180056 A368192 A172506

Adjacent sequences: A012076 A012077 A012078 * A012080 A012081 A012082

Keyword

sign,easy

Author

Patrick Demichel

Extensions

Terms a(13) - a(15) added by Peter Bala, Jan 31 2026

Status

approved