(* a(0)....a(5); a(5) has 19724 decimal digits: *)
a[1, 0, 0] = 1; a[1, 1, 0] = 1; a[1, n_, q_] := 0 /; (q != 0 || (n =!= 0 && n =!= 1)); a[k_Integer?Positive, n_Integer, q_Integer] /; k >= 2 := a[k, n, q] = Sum[Binomial[2^q - 2^(q - m), n] * a[k - 1, m, q - m], {m, 0, q}]; maxq[1] = 0; maxq[k_Integer?Positive] /; 5 >= k >= 2 := maxq[k] = 2^(maxq[k - 1]); aTot[0] = 1; aTotal[k_Integer?Positive] := aTot[k] = Sum[a[k, n, q], {q, 0, maxq[k]}, {n, 0, 2^q}];
Table[aTot[k], {k, 0, 5}] (* or *)
sPrt[mx_, i_, j_] := If[1 <= i <= Length[mx] && 1 <= j <= Length[mx[[i]]], mx[[i, j]], 0]; nxtL[prev_] := Block[{qmax}, qmax = 2^(Length[prev] - 1); Table[ Table[ Sum[ sPrt[prev, q - m + 1, m + 1] * Binomial[2^q - 2^(q - m), n], {m, 0, q}], {n, 0, 2^q}], {q, 0, qmax}]]; aTot[n_] := Prepend[Map[Total[Flatten[#]] &, NestList[nxtL, {{1, 1}}, n]], 1]; Table[aTot[i], {i, 0, 5}] (* _Mikk Heidemaa_, Mar 17 2025 *)
Attachments:
