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MATHEMATICA
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nn = 16; f[list_] :=
Table[Sum[list[[i, j]]*Binomial[i, j], {j, 1, i}] + list[[i, 1]], {i,
1, Length[list]}];
a[x_] := Sum[2^Binomial[n, 2] x^n/n!, {n, 0, nn + 100}];
c[x_] := Log[a[x]]; Prepend[
f[Table[Table[
PadLeft[Range[0, nn]! CoefficientList[
Series[ D[c[ x], {x, n}] a[x], {x, 0, nn}], x], nn + n], {n,
1, nn}][[All, j]], {j, 1, nn}]], 1]
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