A185324 E.g.f. log(1/(2-tan(x)-sec(x))).

0, 1, 2, 7, 34, 215, 1682, 15727, 171274, 2130275, 29799722, 463123747, 7916886514, 147635940335, 2982555226562, 64888568231767, 1512552803481754, 37608099684426395, 993530210286226202, 27791008680163167787, 820556749933610580994, 25502885614554196884455

Offset: 0

Author

Vladimir Kruchinin, Feb 17 2011

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200

Programs

Maple

T:= proc(n,k) option remember;
      if k=n then 1
    elif k<0 or k>n then 0
    else T(n-1, k-1) +k*T(n-1,k) +k*(k+1)/2 *T(n-1, k+1)
      fi
    end:
a:= n-> add((k-1)! * T(n,k), k=1..n):
seq(a(n), n=0..20);  # Alois P. Heinz, Feb 17 2011

Mathematica

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 *)

Maxima

a[0]:0$a[1]:1$
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;
makelist(a[n],n,0,100); /* Tani Akinari, Oct 30 2017 */

Formula

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

a(n) ~ (n-1)! / (arctan(3/4))^n. - Vaclav Kotesovec, Aug 22 2014