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EXAMPLE
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a(0)=1=card{[e]})
a(1)=4=card{[e],[1],[12],[123]})
a(2) = 35 = card({ [e,e],[e,1],[e,12],[1,2],[e,123],[1,23],[2,13],[3,12],
[1,234],[2,134],[3,124],[4,123],[12,34],[13,24],[14,23],[12,345],[13,245],[14,235],[15,324],[23,145],[24,135],[25,134],[34,125],[35,124],[45,123],
[123,456],[124,356],[125,346],[126,345],[134,256],[135,246],[136,245],[145,236],[146,235],[156,234] }) where e denotes the empty set.
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MATHEMATICA
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Table[Sum[k!/(i!3^(i - j)2^(k + j - 2i))Binomial[i, j] Binomial[j, k + 2j - 3i], {k, 0, 3n}, {i, 0, n}, {j, 0, 3i - k}], {n, 0, 15}]
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