%I #4 Jan 19 2013 08:29:46

%S 1,1,2,27,968,68865,8252496,1505652267,390426582272,136910626544673,

%T 62531921536979200,36122742294179711643,25777397243775426776064,

%U 22288717300246130379501921,22978204666564567674247942144,27861330789200983137890612877675

%N E.g.f.: Sum_{n>=0} a(n) * (cos(n*x) - sin(n*x))^n * x^n/n! = 1 + x.

%e By definition, the coefficients a(n) satisfy:

%e 1+x = 1 + 1*(cos(x)-sin(x))*x + 2*(cos(2*x)-sin(2*x))^2*x^2/2! + 27*(cos(3*x)-sin(3*x))^3*x^3/3! + 968*(cos(4*x)-sin(4*x))^4*x^4/4! + 68865*(cos(5*x)-sin(5*x))^5*x^5/5! +...+ a(n)*(cos(n*x)-sin(n*x))^n*x^n/n! +...

%o (PARI) {a(n)=local(A=[1, 1], N); for(i=1, n, A=concat(A, 0); N=#A; A[N]=(N-1)!*(-Vec(sum(m=0, N-1, A[m+1]*x^m/m!*(cos(m*x+x*O(x^N))-sin(m*x+x*O(x^N)))^m))[N])); A[n+1]}

%o for(n=0, 25, print1(a(n), ", "))

%Y Cf. A218798, A221535.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jan 19 2013