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Expansion of e.g.f. cos(x/cos(x)) (even powers only).
7

%I #20 Jul 26 2018 03:21:56

%S 1,-1,-11,-181,-3863,-66121,4478365,1211701763,226423491793,

%T 43068302925551,8876725117679941,1997577117130009403,

%U 483811389670392875449,121594250947356501211559,28960468994349845642813677

%N Expansion of e.g.f. cos(x/cos(x)) (even powers only).

%H G. C. Greubel, <a href="/A009118/b009118.txt">Table of n, a(n) for n = 0..240</a>

%F a(n) = 2*Sum_{m=1..n-1} binomial(2*n,2*m)*Sum_{j=0..n-m} binomial(m+j-1,j)*4^(n-m-j)*Sum_{i=0..j} (i-j)^(2*n-2*m)*binomial(2*j,i)*(-1)^(n+j-i) +(-1)^n. - _Vladimir Kruchinin_, Jun 28 2011

%t With[{nmax = 60}, CoefficientList[Series[Cos[x/Cos[x]], {x, 0, nmax}], x]*Range[0, nmax]!][[1 ;; -1 ;; 2]] (* _G. C. Greubel_, Jul 26 2018 *)

%o (Maxima)

%o a(n):=2*sum(binomial(2*n, 2*m)*sum(binomial(m+j-1, j)*4^(n-m-j)*sum((i-j)^(2*n-2*m)*binomial(2*j, i)*(-1)^(n+j-i), i, 0, j), j, 0, n-m), m, 1, n-1)+(-1)^n; /* _Vladimir Kruchinin_, Jun 28 2011 */

%o (PARI) x='x+O('x^50); v=Vec(serlaplace(cos(x/cos(x)))); vector(#v\2,n,v[2*n-1]) \\ _G. C. Greubel_, Jul 26 2018

%K sign

%O 0,3

%A _R. H. Hardin_

%E Extended with signs by _Olivier GĂ©rard_, Mar 15 1997