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A294035 a(n) = 3^n*hypergeom([-n/3, (1-n)/3, (2-n)/3], [1, 1], -1).

%I A294035 #16 Aug 06 2018 05:34:26
%S A294035 1,3,9,33,153,783,4059,21087,110889,592899,3214989,17608077,97150491,
%T A294035 539331237,3010588317,16887545793,95134584969,537942476907,
%U A294035 3051902823849,17365639042449,99076018204413,566622950463099,3247670747106927,18651711493531539,107315246617831179
%N A294035 a(n) = 3^n*hypergeom([-n/3, (1-n)/3, (2-n)/3], [1, 1], -1).
%C A294035 Diagonal of rational function 1/(1 - (x^3 + y^3 + z^3 + 3*x*y*z)). - _Gheorghe Coserea_, Aug 04 2018
%H A294035 Robert Israel, <a href="/A294035/b294035.txt">Table of n, a(n) for n = 0..1288</a>
%F A294035 Let H(m, n, x) = m^n*hypergeom([(k-n)/m for k=0..m-1], [1 for k=0..m-2], x) then a(n) = H(3, n, -1).
%F A294035 a(n) ~ sqrt(3) * 6^n / (Pi*n) . - _Vaclav Kotesovec_, Nov 02 2017
%F A294035 -(54*(n+2))*(n+1)*a(n)+27*(n+2)^2*a(n+1)-(3*(3*n^2+15*n+19))*a(n+2)+(n+3)^2*a(n+3) = 0. - _Robert Israel_, Nov 02 2017
%p A294035 T := (m,n,x) -> m^n*hypergeom([seq((k-n)/m, k=0..m-1)], [seq(1,k=0..m-2)], x):
%p A294035 seq(simplify(T(3, n, -1)), n=0..39);
%t A294035 Table[3^n * HypergeometricPFQ[{-n/3, (1 - n)/3, (2 - n)/3}, {1, 1}, -1], {n, 0, 30}] (* _Vaclav Kotesovec_, Nov 02 2017 *)
%Y A294035 H(1, n, 1) = A000007(n), H(2, n, 1) = A000984(n), H(3, n, 1) = A006077(n), H(4, n, 1) = A294036(n), H(1, n, -1) = A000079(n), H(2, n, -1) = A098335(n), H(3, n, -1) = this seq., H(4, n, -1) = A294037(n).
%K A294035 nonn
%O A294035 0,2
%A A294035 _Peter Luschny_, Nov 02 2017