A317882 Number of free pure achiral multifunctions (with empty expressions allowed) with one atom and n positions.

1, 1, 2, 5, 12, 31, 79, 211, 564, 1543, 4259, 11899, 33526, 95272, 272544, 784598, 2270888, 6604900, 19293793, 56581857, 166523462, 491674696, 1455996925, 4323328548, 12869353254, 38396655023, 114803257039, 343932660450, 1032266513328, 3103532577722

Offset

1,3

Comments

A free pure achiral multifunction (with empty expressions allowed) (AME) is either (case 1) the leaf symbol "o", or (case 2) a possibly empty expression of the form h[g, ..., g] where h and g are AMEs. The number of positions in an AME is the number of brackets [...] plus the number of o's.

Also the number of achiral Mathematica expressions with one atom and n positions.

Links

Andrew Howroyd, Table of n, a(n) for n = 1..200

Formula

a(1) = 1; a(n > 1) = a(n - 1) + Sum_{0 < k < n - 1} a(k) * Sum_{d|(n - k - 1)} a(d).

Example

The a(5) = 12 AMEs:
  o[o[o]]
  o[o][o]
  o[o[][]]
  o[o,o,o]
  o[][o[]]
  o[][o,o]
  o[][][o]
  o[o[]][]
  o[o,o][]
  o[][o][]
  o[o][][]
  o[][][][]

Mathematica

a[n_]:=If[n==1, 1, Sum[a[k]*If[k==n-1, 1, Sum[a[d], {d, Divisors[n-k-1]}]], {k, n-1}]];
Array[a, 12]

Prog

(PARI) seq(n)={my(p=O(x)); for(n=1, n, p = x + p*x*(1 + sum(k=1, n-2, subst(p + O(x^(n\k+1)), x, x^k)) ) + O(x*x^n)); Vec(p)} \\ Andrew Howroyd, Aug 19 2018
(PARI) seq(n)={my(v=vector(n)); v[1]=1; for(n=2, #v, v[n]=v[n-1] + sum(i=1, n-2, v[i]*sumdiv(n-i-1, d, v[d]))); v} \\ Andrew Howroyd, Aug 19 2018

Crossrefs

Cf. A001003, A002033, A003238, A052893, A053492, A214577, A277996, A317853, A317875.

Cf. A317883, A317884, A317885.

Sequence in context: A238829 A125023 A129804 * A335457 A290616 A110035

Adjacent sequences: A317879 A317880 A317881 * A317883 A317884 A317885

Keyword

nonn

Author

Gus Wiseman, Aug 09 2018

Extensions

Terms a(13) and beyond from Andrew Howroyd, Aug 19 2018

Status

approved