A317885 Number of series-reduced free pure achiral multifunctions with one atom and n positions.

1, 0, 0, 1, 1, 1, 2, 3, 4, 7, 9, 14, 21, 32, 45, 69, 103, 153, 224, 338, 500, 746, 1107, 1645, 2447, 3652, 5413, 8052, 11993, 17834, 26500, 39447, 58655, 87240, 129772, 193001, 287034, 427014, 635048, 944501, 1404910, 2089633, 3107864, 4622670, 6875533

Offset

1,7

Comments

A series-reduced free pure achiral multifunction (SRAM) is either (case 1) the leaf symbol "o", or (case 2) a nonempty and non-unitary expression of the form h[g, ..., g] where h and g are SRAMs. The number of positions in a SRAM is the number of brackets [...] plus the number of o's.

Links

Andrew Howroyd, Table of n, a(n) for n = 1..500

Formula

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

Example

The a(10) = 7 SRAMs:
  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,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]*Sum[a[d], {d, Most[Divisors[n-k-1]]}], {k, n-2}]];
Array[a, 12]

Prog

(PARI) seq(n)={my(p=O(x)); for(n=1, n, p = x + p*x*sum(k=2, 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]=sum(i=1, n-2, v[i]*sumdiv(n-i-1, d, if(d<n-i-1, v[d], 0)))); v} \\ Andrew Howroyd, Aug 19 2018

Crossrefs

Cf. A001003, A001678, A002033, A003238, A052893, A053492, A067824, A167865, A214577, A317853, A317875.

Cf. A317882, A317883, A317884.

Sequence in context: A094093 A240077 A237997 * A321535 A108809 A276846

Adjacent sequences: A317882 A317883 A317884 * A317886 A317887 A317888

Keyword

nonn

Author

Gus Wiseman, Aug 09 2018

Extensions

Terms a(17) and beyond from Andrew Howroyd, Aug 19 2018

Status

approved