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a(n) = Sum_{k=0..n} T(n,k) * T(n,n+k), with T given by A027082.
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%I #13 Sep 26 2020 11:41:17

%S 1,2,6,29,131,619,3013,14854,73996,372023,1883369,9591012,49091034,

%T 252371780,1302419484,6744389329,35031370911,182457629783,

%U 952674694705,4985494458892,26143834054498,137358293523986,722938114992478

%N a(n) = Sum_{k=0..n} T(n,k) * T(n,n+k), with T given by A027082.

%t T[_, 0] = T[_, 1] = T[_, 2] = 1;

%t T[n_, k_] := T[n, k] = If[k < 2n, T[n-1, k-3] + T[n-1, k-2] + T[n-1, k-1], T[n-1, 2n-3] + T[n-1, 2n-2]];

%t a[n_] := Sum[T[n, k] T[n, n+k], {k, 0, n}];

%t a /@ Range[0, 22] (* _Jean-François Alcover_, Sep 26 2020 *)

%Y Cf. A027082.

%K nonn

%O 0,2

%A _Clark Kimberling_