OFFSET
0,3
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..240
FORMULA
E.g.f.: exp(sec(x)-1) = 1 + Sum_{n>0} a(n)*x^(2*n)/(2*n)!.
a(n) = Sum_{m=1..(2*n)} (Sum_{k=m..(2*n)} binomial(k-1,m-1)*Sum_{j=(2*k)..(2*n)} binomial(j-1,2*k-1)*(j)!*2^(m-j)*(-1)^(n+k+j) * stirling2(2*n,j))))/m!), n>0, a(0)=1.
a(n) ~ 2^(4*n+1/2)* exp(4*sqrt(n)/sqrt(Pi)-2*n-1+1/Pi)* n^(2*n-1/4) / Pi^(2*n+1/4). - Vaclav Kotesovec, Sep 24 2013
a(n) = Sum_{i=0..(n-1)} binomial(2*n-1,2*i+1)*z(i)*a(n-i-1))), a(0)=1, where z(n) = euler(n+1) - secant numbers (A000364). - Vladimir Kruchinin, Mar 01 2015
MAPLE
a := proc(n) option remember; if n=0 then 1 else add((-1)^i*binomial(2*n-1, 2*i-1)* euler(2*i)*a(n-i), i=1..n) fi end: seq(a(n), n=0..15); # Peter Luschny, Mar 08 2015
MATHEMATICA
a[n_] := Sum[ (Sum[ Binomial[k - 1, m - 1]* Sum[ Binomial[j - 1, 2*k - 1]*(j)!*2^(m - j)*(-1)^(n + k + j)*StirlingS2[2*n, j], {j, 2*k, 2*n}], {k, m, 2*n}])/m!, {m, 1, 2*n}]; a[0] = 1; Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Feb 22 2013 *)
Table[(2*n)!*SeriesCoefficient[E^(1/Cos[x]-1), {x, 0, 2*n}], {n, 0, 20}] (* Vaclav Kotesovec, Sep 24 2013 *)
With[{nn = 100}, CoefficientList[Series[Exp[Sec[x] - 1], {x, 0, nn}],
x] Range[0, nn]!][[;; ;; 2]] (* G. C. Greubel, May 31 2017 *)
PROG
(Maxima) a(n):=if n=0 then 1 else sum((sum(binomial(k-1, m-1)*sum(binomial(j-1, 2*k-1)*(j)!*2^(m-j)*(-1)^(n+k+j)*stirling2(2*n, j), j, 2*k, 2*n), k, m, 2*n))/m!, m, 1, 2*n);
(Maxima)
z(n):=(2*n+2)!*(coeff(taylor(sec(x), x, 0, 20), x, 2*(n+1)));
a(n):=if n=0 then 1 else (sum(binomial(2*n-1, 2*i+1)*z(i)*a(n-i-1), i, 0, n-1)); /* Vladimir Kruchinin, Mar 01 2015 */
(PARI) a(n) = {n = 2*n+2; xx = x + O(x^n); polcoeff(serlaplace(exp(1/cos(xx)-1)), n); } \\ Michel Marcus, Mar 03 2015
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladimir Kruchinin, Oct 05 2012
STATUS
approved