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%I #16 Oct 09 2020 05:17:38
%S 0,1,1,2,4,6,14,21,49,77,175,287,637,1079,2353,4082,8788,15522,33098,
%T 59280,125476,227240,478192,873886,1830270,3370030,7030570,13027730,
%U 27088870,50469890,104647630,195892565,405187825,761615285,1571990935,2965576715
%N a(n) = Sum_{k=1..n-1} ((-1)^(k+n+1)*binomial(k,floor(k/2))).
%C a(n) is the sum of the alternating series of central binomial coefficients (including all rows, defined as binomial(m,floor(m/2)) or equivalently binomial(m,ceiling(m/2)) for all m odd, A001405).
%H Robert Israel, <a href="/A332754/b332754.txt">Table of n, a(n) for n = 1..3326</a>
%F a(n) = Sum_{k=1..n-1} ((-1)^(k+n+1)*binomial(k,floor(k/2))).
%F From _Robert Israel_, Oct 08 2020: (Start)
%F D-finite with recurrence: (4*n - 8)*a(n - 3) + (-6 + 4*n)*a(n - 2) + (-n + 2)*a(n - 1) - n*a(n) = 0.
%F G.f. (sqrt((1+2*x)/(1-2*x))-1-2*x)/(2+2*x). (End)
%p f:= gfun:-rectoproc({(4*n + 4)*a(n) + (6 + 4*n)*a(n + 1) + (-n - 1)*a(n + 2) + (-n - 3)*a(n + 3), a(0) = 0, a(1) = 0, a(2) = 1, a(3) = 1},a(n),remember):
%p map(f, [$1..100]); # _Robert Israel_, Oct 08 2020
%t Sum[(-1)^(k + n + 1) Binomial[k, Floor[k/2]], {k, 1, -1 + n}]
%o (PARI) a(n) = sum(k=1, n-1, (-1)^(k+n+1)*binomial(k, k\2)); \\ _Michel Marcus_, Feb 22 2020
%K nonn
%O 1,4
%A _Brian P Hawkins_, Feb 22 2020