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A317883
Number of free pure achiral multifunctions with one atom and n positions.
6
1, 0, 1, 1, 3, 4, 10, 17, 37, 70, 150, 299, 634, 1311, 2786, 5879, 12584, 26904, 58005, 125242, 271819, 591297, 1290976, 2825170, 6199964, 13635749, 30057649, 66386206, 146903289, 325637240, 723024160, 1607805207, 3580476340, 7984266625, 17827226469
OFFSET
1,5
COMMENTS
A free pure achiral multifunction (PAM) is either (case 1) the leaf symbol "o", or (case 2) a nonempty expression of the form h[g, ..., g] where h and g are PAMs. The number of positions in a PAM is the number of brackets [...] plus the number of o's.
LINKS
FORMULA
a(1) = 1; a(n > 1) = Sum_{0 < k < n - 1} a(k) * Sum_{d|(n - k - 1)} a(d).
G.f. A(x) satisfies: A(x) = x * (1 + A(x) * Sum_{k>=1} A(x^k)). - Ilya Gutkovskiy, May 03 2019
EXAMPLE
The a(7) = 10 PAMs:
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, 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=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]=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