A217502 - OEIS

%I #32 Apr 14 2023 16:26:27

%S 1,1,8,151,5123,271396,20605133,2116186801,282013329788,

%T 47257934281891,9716069206252163,2402866414155189016,

%U 703288162788887287433,240323111593250975343601,94776477297941909597367248,42710529437686482677512782271

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

%H G. C. Greubel, <a href="/A217502/b217502.txt">Table of n, a(n) for n = 0..240</a>

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

%F 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.

%F 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 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 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 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 Table[(2*n)!*SeriesCoefficient[E^(1/Cos[x]-1), {x,0,2*n}], {n,0,20}] (* _Vaclav Kotesovec_, Sep 24 2013 *)

%t With[{nn = 100}, CoefficientList[Series[Exp[Sec[x] - 1], {x, 0, nn}],

%t x] Range[0, nn]!][[;; ;; 2]] (* _G. C. Greubel_, May 31 2017 *)

%o (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 (Maxima)

%o z(n):=(2*n+2)!*(coeff(taylor(sec(x),x,0,20),x,2*(n+1)));

%o 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 (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 Cf. A000364.

%K nonn

%O 0,3

%A _Vladimir Kruchinin_, Oct 05 2012