OFFSET
0,2
LINKS
T. D. Noe, Table of n, a(n) for n = 0..100
FORMULA
G.f.: 1/G(0) - 1/(1+x) where G(k) = 1 - x*(2*k+1)^2/(1 - x*(2*k+2)^2/G(k+1) ); (recursively defined continued fraction). - Sergei N. Gladkovskii, Feb 05 2013
G.f.: 1/G(0) - 1/(1+x) where G(k) = 1 - x*(k+1)^2/G(k+1); (recursively defined continued fraction). - Sergei N. Gladkovskii, Feb 09 2013
a(n) ~ (2*n)! * 4^(n+1) / Pi^(2*n+1). - Vaclav Kotesovec, Jan 24 2015
Conjectural o.g.f.: Sum_{n >= 0} 4*x/2^n * Sum_{k = 0..n} (-1)^k*(k+1)*binomial(n,k)/( (1 + x*(2*k + 1)^2)*(1 + x*(2*k + 3)^2) ) = 2*x + 4*x^2 + 62*x^3 + 1384*x^4 + .... - Peter Bala, Mar 03 2015
From Peter Luschny, Jun 13 2021: (Start)
a(n) = (-1)^n*(Euler(2*n) - 1).
a(n) ~ 4^(2*n + 3/2)*exp(1/(24*n) - 2*n)*(n/Pi)^(2*n + 1/2). (End)
MAPLE
seq((2*i)!*coeff(series(tan(x)*sin(x), x, 30), x, 2*i), i=0..14); # Peter Luschny, Jul 14 2012
MATHEMATICA
nn = 30; t = Range[0, nn]! CoefficientList[Series[Tan[x]*Sin[x], {x, 0, nn}], x]; Take[t, {1, nn, 2}] (* T. D. Noe, Jul 15 2012 *)
PROG
(Sage) # Variant of an algorithm of L. Seidel (1877) with a(0) = 1.
def A009744_list(n) :
dim = 2*n; E = matrix(ZZ, dim); E[0, 0] = 1
for m in (1..dim-1) :
if m % 2 == 0 :
E[m, 0] = 1;
for k in range(m-1, -1, -1) :
E[k, m-k] = E[k+1, m-k-1] - E[k, m-k-1]
else :
E[0, m] = 1;
for k in range(1, m+1, 1) :
E[k, m-k] = E[k-1, m-k+1] + E[k-1, m-k]
return [(-1)^(k//2)*E[0, k] for k in range(dim) if is_even(k)]
A009744_list(14) # Peter Luschny, Jul 14 2012
(PARI) x='x+O('x^50); v=Vec(serlaplace(tan(x)*sin(x))); concat([0], vector(#v\2, n, v[2*n-1])) \\ G. C. Greubel, Mar 04 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Extended and signs tested by Olivier Gérard, Mar 15 1997
STATUS
approved