The Wolfram code to compute the sequence A381080(5) ...

Ideas
1

(* 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 *)

A381080.nb (47.5 KB)

2

… a(5) has 19724 decimal digits.

3

Would like to know its factors / divisors… :wink:

4

Abelian orders of hereditarily finite transitive sets.

Sequence:
1, 2, 3, 6, 12, 2^13868 * 3^7873 * 5^2012 * 7^1911, …

Conjecture: a(7) = 2^a ⋅ 3^b ⋅ 5^c ⋅ 7^d ⋅ 11^e ⋅13^f ⋅ … ⋅ P^z.

Try to prove / disprove… (or to find a(7)…?)

5

Mikk Heidemaa, A Formal Presentation of Two Sequences, Their Gödel-Like Encoding, and a Commutative Group-Theoretic Result, (arXiv submission 6339673) (2025), 1-6.
(A381080 - OEIS)