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COMMENT
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A unit column of a binary matrix is a column with only one 1. First differences of a(n) give number of minimal 4-covers of an unlabeled n-set that cover 5 points of that set uniquely (if offset is 5).
Generally, the number b(n, k) of 4 X n binary matrices with k=0, 1, ..., n unit columns, up to row and column permutations, is coefficient of x^k in 1/24*(Z(S_n; 12 + 4*x, 12 + 4*x^2, ... ) + 8*Z(S_n; 3 + x, 3 + x^2, 12 + 4*x^3, 3 + x^4, 3 + x^5, 12 + 4*x^6, ...) + 6*Z(S_n; 6 + 2*x, 12 + 4*x^2, 6 + 2*x^3, 12 + 4*x^4, ...) + 3*Z(S_n; 4, 12 + 4*x^2, 4, 12 + 4*x^4, ...) + 6*Z(S_n; 2, 4, 2, 12 + 4*x^4, 2, 4, 2, 12 + 4*x^8, ...)), where Z(S_n; x_1, x_2, ..., x_n) is cycle index of symmetric group S_n of degree n.
G.f. for b(n,k), k=0,1,..,n, is 1/k!* k - th derivative of 1/24*(1/(1 - x)^12/(1 - x*t)^4 + 8/(1 - x)^3/(1 - x^3)^3/(1 - x^3*t^3)/(1 - x*t) + 6/(1 - x)^6/(1 - x^2)^3/(1 - x^2*t^2)/(1 - x*t)^2 + 3/(1 - x)^4/(1 - x^2)^4/(1 - x^2*t^2)^2 + 6/(1 - x)^2/(1 - x^2)/(1 - x^4)^2/(1 - x^4*t^4)) with respect to t, when t=0.
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