A185324 - OEIS

%I #30 Dec 26 2023 10:13:52

%S 0,1,2,7,34,215,1682,15727,171274,2130275,29799722,463123747,

%T 7916886514,147635940335,2982555226562,64888568231767,

%U 1512552803481754,37608099684426395,993530210286226202,27791008680163167787,820556749933610580994,25502885614554196884455

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

%H Alois P. Heinz, <a href="/A185324/b185324.txt">Table of n, a(n) for n = 0..200</a>

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

%F a(n) ~ (n-1)! / (arctan(3/4))^n. - _Vaclav Kotesovec_, Aug 22 2014

%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-1)! * T(n,k), k=1..n):

%p seq(a(n), n=0..20);  # _Alois P. Heinz_, Feb 17 2011

%t T[n_, k_] := T[n, k] = If[k==n, 1, If[k<0 || k>n, 0, T[n-1, k-1] + k*T[n-1, k] +  k*(k+1)/2*T[n-1, k+1]]]; a[n_] := Sum[(k-1)!*T[n, k], {k, 1, n}]; Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Apr 03 2015, after _Alois P. Heinz_ *)

%o (Maxima) a[0]:0$a[1]:1$

%o a[n]:=sum((-1)^floor(p/2)*(mod(p+1,2)-(-1)^p*4^floor(p/2))*binomial(n-1,p)*a[n-p],p,1,n-1)-mod(n-1,2)*(%i)^n;

%o makelist(a[n],n,0,100); /* _Tani Akinari_, Oct 30 2017 */

%K nonn

%O 0,3

%A _Vladimir Kruchinin_, Feb 17 2011