OFFSET
0,2
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..200
Vladimir Kruchinin, The method for obtaining expressions for coefficients of reverse generating functions, arXiv:1211.3244 [math.CO], 2012.
FORMULA
a(n) = 2*Sum_{k=1..2*n} binomial(2*n+k,2*n)*(Sum_{j=1..k} ((Sum_{i=0..(j-1)/2)} (j-2*i)^(2*n)*binomial(j,i))*binomial(k,j)*(-1)^(n-j))/2^j))), n>0, a(0)=1.
a(n) = [x^(2*n)/(2*n)!] 1/cos(x)^(2*n+1). - Paul D. Hanna, Jan 23 2012
a(n) = (2*n+1) * A196873(n) for n>=1, where e.g.f. G(x) of A196873 satisfies: G(x*cos(x)) = 1/cos(x). - Paul D. Hanna, Jan 23 2012
a(n) = Sum_{k=1..2*n} (binomial(2*n+k,2*n)*Sum_{i=0..k-1} (i-k)^(2*n)*binomial(2*k,i)*(-1)^(n+k-i)))/2^(k-1), with n>0, a(0)=1. - Vladimir Kruchinin, Oct 08 2012
MATHEMATICA
a[n_] := Sum[ (Binomial[2*n + k, 2*n]*Sum[ (i - k)^(2*n)*Binomial[2*k, i]*(-1)^(n + k - i), {i, 0, k - 1}])/2^(k - 1), {k, 1, 2*n}]; a[0] = 1; Table[a[n], {n, 0, 12}] (* Jean-François Alcover, Feb 21 2013, translated from Maxima *)
PROG
(Maxima) a(n):=if n=0 then 1 else 2*sum(binomial(2*n+k, 2*n)*sum(((sum((j-2*i)^(2*n)*binomial(j, i), i, 0, (j-1)/2))*binomial(k, j)*(-1)^(n-j))/2^j, j, 1, k), k, 1, 2*n)/(2*n+1)!;
(PARI) {a(n)=if(n==0, 1, 2*sum(k=1, 2*n, binomial(2*n+k, 2*n)*sum(j=1, k, sum(i=0, floor((j-1)/2), (j-2*i)^(2*n)*binomial(j, i))*binomial(k, j)*(-1)^(n-j)/2^j)))}
(PARI) {a(n)=(2*n)!*polcoeff(1/cos(x+x*O(x^(2*n+1)))^(2*n+1), 2*n)}
(Maxima) a(n):=if n=0 then 1 else (sum((binomial(2*n+k, 2*n)*sum((i-k)^(2*n)*binomial(2*k, i)*(-1)^(n+k-i), i, 0, k-1))/2^(k-1), k, 1, 2*n));
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladimir Kruchinin, Jan 23 2012
STATUS
approved