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A317882
Number of free pure achiral multifunctions (with empty expressions allowed) with one atom and n positions.
6
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
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
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 09 2018
EXTENSIONS
Terms a(13) and beyond from Andrew Howroyd, Aug 19 2018
STATUS
approved