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A217502 E.g.f.: exp(sec(x)-1) = 1 + Sum_{n>0} a(n)*x^(2*n)/(2*n)!.

%I A217502 #37 May 15 2025 05:28:11
%S A217502 1,1,8,151,5123,271396,20605133,2116186801,282013329788,
%T A217502 47257934281891,9716069206252163,2402866414155189016,
%U A217502 703288162788887287433,240323111593250975343601,94776477297941909597367248,42710529437686482677512782271
%N A217502 E.g.f.: exp(sec(x)-1) = 1 + Sum_{n>0} a(n)*x^(2*n)/(2*n)!.
%H A217502 G. C. Greubel, <a href="/A217502/b217502.txt">Table of n, a(n) for n = 0..240</a>
%F A217502 E.g.f.: exp(sec(x)-1) = 1 + Sum_{n>0} a(n)*x^(2*n)/(2*n)!.
%F A217502 a(n) = Sum_{m=1..(2*n)} (1/m!)*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), n>0, a(0)=1.
%F A217502 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
%F A217502 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
%p A217502 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
%t A217502 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 *)
%t A217502 Table[(2*n)!*SeriesCoefficient[E^(1/Cos[x]-1), {x,0,2*n}], {n,0,20}] (* _Vaclav Kotesovec_, Sep 24 2013 *)
%t A217502 With[{nn = 100}, CoefficientList[Series[Exp[Sec[x] - 1], {x, 0, nn}],
%t A217502 x] Range[0, nn]!][[;; ;; 2]] (* _G. C. Greubel_, May 31 2017 *)
%o A217502 (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);
%o A217502 (Maxima)
%o A217502 z(n):=(2*n+2)!*(coeff(taylor(sec(x),x,0,20),x,2*(n+1)));
%o A217502 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 */
%o A217502 (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
%Y A217502 Cf. A000364.
%K A217502 nonn
%O A217502 0,3
%A A217502 _Vladimir Kruchinin_, Oct 05 2012