A155585 Bernoulli numbers matrix inverse sum

nn = 20;
A = Table[
  Table[If[n >= k, Binomial[n - 1, k - 1], 0]*
    If[And[n > k, Mod[k, 2] == 0], 0, 1], {k, 1, nn}], {n, 1,
   nn}]; B =
 Table[Table[
   If[Mod[n - k, 2] == 1, 0, 1]*If[n >= k, Binomial[n - 1, k - 1], 0]*
    If[And[n > k, Mod[k, 2] == 0], 0, 1], {k, 1, nn}], {n, 1, nn}];
W = Inverse[B] - Inverse[A] + IdentityMatrix[nn];
Inverse[B][[All, 1]]
Inverse[A][[All, 1]]
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https : // oeis.org/A155585