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O.g.f.:G(G(x)-1)where G(x)=Sum_k>=0,k!*x^k
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%I #2 Mar 30 2012 17:23:10

%S 1,1,4,20,116,756,5448,43032,370704,3472656,35303424,388786176,

%T 4628774016,59431077504,820503839232,12140411779584,191844739736064,

%U 3226219598338560,57540789020021760,1084906603389864960

%N O.g.f.:G(G(x)-1)where G(x)=Sum_k>=0,k!*x^k

%C a(n) is the number of ways to create(select) each Ferrers "type" diagram of the ORDERED partitions (compositions), then label (linearly order) the cells within each row of the diagram, and then linearly order each row.

%e a(3)=20 because each composition of three: 3,2+1,1+2,1+1+1 can be so ordered in 3!+2!*2!+2!*2!+3!=20 ways respectively.

%t A[x_] = Sum[i! x^i, {i, 1, 20}] B[x_] = Sum[i! x^i, {i, 0, 20}] CoefficientList[Expand[Composition[B, A][x]], x]

%K nonn

%O 0,3

%A _Geoffrey Critzer_, Mar 29 2009