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A187916 a(n) = C(n) if n is odd, else C(n) - C(n/2); C(n) are Catalan numbers.

%I A187916 #8 Jul 21 2015 05:46:26
%S A187916 1,1,5,12,42,127,429,1416,4862,16754,58786,207880,742900,2674011,
%T A187916 9694845,35356240,129644790,477633838,1767263190,6564103624,
%U A187916 24466267020,91482504854,343059613650,1289903939312,4861946401452,18367352329252,69533550916004
%N A187916 a(n) = C(n) if n is odd, else C(n) - C(n/2); C(n) are Catalan numbers.
%F A187916 Conjecture: -(n-2)*(n+2)*(n+1)*a(n) +8*(n+1)*(n^2-3*n+1)*a(n-1) +4*(-3*n^3+18*n^2-22*n+2)*a(n-2) -32*(n-2)*(n^2-3*n+1)*a(n-3) +16*
%F A187916 (2*n-7)*(n-3)*(2*n-3)*a(n-4)=0. - _R. J. Mathar_, Jul 21 2015
%e A187916 a(6) = 127 = A000108(6) - A000108(3) = (132 - 5)
%e A187916 a(5) = 42 = A000108(5)
%p A187916 A187916 := proc(n)
%p A187916     if type(n,'odd') then
%p A187916         A000108(n);
%p A187916     else
%p A187916         A000108(n)-A000108(n/2);
%p A187916     end if;
%p A187916 end proc:
%p A187916 seq(A187916(n),n=1..30) ; # _R. J. Mathar_, Jul 21 2015
%t A187916 Table[If[OddQ[n],CatalanNumber[n],CatalanNumber[n]-CatalanNumber[n/2]],{n,30}] (* _Harvey P. Dale_, May 22 2014 *)
%Y A187916 Cf. A000912, A000108
%K A187916 nonn,easy
%O A187916 1,3
%A A187916 _Gary W. Adamson_, Mar 15 2011