A352151 Expansion of e.g.f. 1/(cos(x) - tan(x)).
1, 1, 3, 14, 81, 616, 5523, 58064, 697281, 9417856, 141368643, 2334020864, 42039523281, 820296426496, 17237259945363, 388087200241664, 9320064293358081, 237814050877505536, 6425096888209255683, 183232685725482942464, 5500505587921088841681
Offset: 0
Keywords
Programs
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Mathematica
m = 20; Range[0, m]! * CoefficientList[Series[1/(Cos[x] - Tan[x]), {x, 0, m}], x] (* Amiram Eldar, Mar 06 2022 *) -
PARI
my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(cos(x)-tan(x)))) -
PARI
c(n) = ((-4)^n-(-16)^n)*bernfrac(2*n)/(2*n); b(n) = if(n%2==1, c((n+1)/2), (-1)^(n/2+1)); a(n) = if(n==0, 1, sum(k=1, n, b(k)*binomial(n, k)*a(n-k)));
Formula
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).
From Vaclav Kotesovec, Mar 06 2022: (Start)
a(n) ~ n! / (sqrt(5) * (arctan(sqrt((sqrt(5) - 1)/2)))^(n+1)).
a(n) ~ n! / (sqrt(5) * A175288^(n+1)). (End)