A185323 - OEIS

%I #25 Jun 27 2017 06:23:06

%S 1,1,3,14,87,676,6303,68564,852387,11921476,185259603,3166825364,

%T 59054916687,1193026564276,25955467164903,605021502144164,

%U 15043243752072987,397412126087559076,11116403953041202203,328222705791221254964

%N E.g.f. A(x) = 1/(2-tan(x)-sec(x)).

%H G. C. Greubel, <a href="/A185323/b185323.txt">Table of n, a(n) for n = 0..415</a>

%H Vladimir Kruchinin, D. V. Kruchinin, <a href="http://arxiv.org/abs/1103.2582">Composita and their properties</a>, arXiv:1103.2582 [math.CO], 2011.

%F a(n) = Sum_{k=1..n} k!*A147315(n,k), n>0. a(0)=1.

%F E.g.f.: 1 + x/(U(0)-x) where U(k)= 4*k+1 - x/(2 - x/(4*k+3 + x/(2 + x/U(k+1))));(continued fraction, 4-step). - _Sergei N. Gladkovskii_, Nov 08 2012

%F a(n) ~ n! * 2/(5*arctan(3/4)^(n+1)). - _Vaclav Kotesovec_, Sep 25 2013

%p T:= proc(n, k) option remember;

%p       if k=n then 1

%p     elif k<0 or k>n then 0

%p     else T(n-1, k-1) +k*T(n-1, k) +k*(k+1)/2 *T(n-1, k+1)

%p       fi

%p     end:

%p a:= n-> add(k! * T(n, k), k=0..n):

%p seq(a(n), n=0..30);  # _Alois P. Heinz_, Feb 18 2011

%t CoefficientList[Series[1/(2-Tan[x]-Sec[x]), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Sep 25 2013 *)

%o (PARI) x = 'x + O('x^30); Vec(serlaplace(1/(2-tan(x)-1/cos(x)))) \\ _Michel Marcus_, Jun 27 2017

%K nonn

%O 0,3

%A _Vladimir Kruchinin_, Feb 17 2011