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a(n) is the constant term in the expansion of (1 + (1+x)*(1+y) + (1+1/x)*(1+1/y) + (1+1/x^2)*(1+1/y^2))^n.
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%I #4 Oct 28 2019 07:14:33

%S 1,4,22,157,1342,12694,126601,1301353,13647622,145280776,1564974682,

%T 17022150688,186647326849,2060538384454,22880172406939,

%U 255336518921572,2861956399934038,32201882791307904,363561893114484868,4117138703073839926,46751784944876067562

%N a(n) is the constant term in the expansion of (1 + (1+x)*(1+y) + (1+1/x)*(1+1/y) + (1+1/x^2)*(1+1/y^2))^n.

%t F[n_] := Expand[(1+(1+x)(1+y) + (1+1/x)(1+1/y) + (1+1/x^2)(1+1/y^2))^n];

%t a[n_] := SeriesCoefficient[F[n], {x, 0, 0}, {y, 0, 0}]; a /@ Range[0, 20]

%Y Cf. A002893.

%K nonn

%O 0,2

%A _Peter Luschny_, Oct 28 2019