A352151 - OEIS

%I #27 Mar 06 2022 23:09:17

%S 1,1,3,14,81,616,5523,58064,697281,9417856,141368643,2334020864,

%T 42039523281,820296426496,17237259945363,388087200241664,

%U 9320064293358081,237814050877505536,6425096888209255683,183232685725482942464,5500505587921088841681

%N Expansion of e.g.f. 1/(cos(x) - tan(x)).

%F a(0) = 1; a(n) = Sum_{k=1..n} b(k) * binomial(n,k) * a(n-k), where b(k) = A000182((k+1)/2) if k is odd, otherwise (-1)^(k/2+1).

%F From _Vaclav Kotesovec_, Mar 06 2022: (Start)

%F a(n) ~ n! / (sqrt(5) * (arctan(sqrt((sqrt(5) - 1)/2)))^(n+1)).

%F a(n) ~ n! / (sqrt(5) * A175288^(n+1)). (End)

%t m = 20; Range[0, m]! * CoefficientList[Series[1/(Cos[x] - Tan[x]), {x, 0, m}], x] (* _Amiram Eldar_, Mar 06 2022 *)

%o (PARI) my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(cos(x)-tan(x))))

%o (PARI) c(n) = ((-4)^n-(-16)^n)*bernfrac(2*n)/(2*n);

%o b(n) = if(n%2==1, c((n+1)/2), (-1)^(n/2+1));

%o a(n) = if(n==0, 1, sum(k=1, n, b(k)*binomial(n, k)*a(n-k)));

%Y Cf. A000182, A000828, A001586, A175288, A352164.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Mar 06 2022