%I #8 Sep 08 2022 08:45:51
%S 1,-1,-1,0,-1,1,-3,6,-15,36,-91,232,-603,1585,-4213,11298,-30537,
%T 83097,-227475,625992,-1730787,4805595,-13393689,37458330,-105089229,
%U 295673994,-834086421,2358641376,-6684761125,18985057351,-54022715451
%N First column of A174295.
%C First 6 terms as in Mobius function A008683. Signed version of A099323 with an additional leading 1.
%H G. C. Greubel, <a href="/A174297/b174297.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = -(-3)^(n-3/2)*hypergeometric2F1([3/2, n-1],[2],4) for n > 2. - _Mark van Hoeij_, Jul 02 2010
%F a(n) = (-1)^n if n < 2 otherwise Sum_{j=0..n-2} (-1)^(j-1)*binomial(n-2, j)*Catalan(j). - _G. C. Greubel_, Nov 25 2021
%t a[n_]:= a[n]= If[n<2, (-1)^n, Sum[(-1)^(j+1)*Binomial[n-2, j]*CatalanNumber[j], {j, 0, n-2}]]; Table[a[n], {n,0,40}] (* _G. C. Greubel_, Nov 25 2021 *)
%o (Magma)
%o a:= func< n | n lt 2 select (-1)^n else (&+[(-1)^(k+1)*Binomial(n-2, k)*Catalan(k): k in [0..n-2]]) >;
%o [a(n): n in [0..30]]; // _G. C. Greubel_, Nov 25 2021
%o (Sage) [1,-1]+[sum( (-1)^(j+1)*binomial(n-2,j)*catalan_number(j) for j in (0..n-2) ) for n in (2..40)] # _G. C. Greubel_, Nov 25 2021
%Y Cf. 099323, A112467, A112468, A174294, A174295, A174296.
%K sign
%O 0,7
%A _Mats Granvik_, Mar 15 2010