OFFSET
1,3
COMMENTS
Also the number of distinct orderless Mathematica expressions with one atom and n positions.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..200
Mathematica Reference, Orderless.
FORMULA
From Ilya Gutkovskiy, Apr 30 2019: (Start)
G.f. A(x) satisfies: A(x) = x * (1 + A(x) * exp(Sum_{k>=1} A(x^k)/k)).
G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x * (1 + (Sum_{n>=1} a(n)*x^n) * Product_{n>=1} 1/(1 - x^n)^a(n)). (End)
EXAMPLE
The a(5)=13 Mathematica expressions are:
x[x,x,x]
x[x,x][] x[x][x] x[][x,x] x[x,x[]] x[x[x]]
x[x][][] x[][x][] x[][][x] x[x[]][] x[][x[]] x[x[][]]
x[][][][]
MATHEMATICA
multing[t_, n_]:=Array[(t+#-1)/#&, n, 1, Times];
a[n_]:=a[n]=If[n===1, 1, Sum[a[k]*Sum[Product[multing[a[First[s]], Length[s]], {s, Split[p]}], {p, IntegerPartitions[n-k-1]}], {k, 1, n-1}]];
Array[a, 30]
PROG
(PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
seq(n)={my(v=[1]); for(n=2, n, my(t=EulerT(v)); v=concat(v, v[n-1] + sum(k=1, n-2, v[k]*t[n-k-1]))); v} \\ Andrew Howroyd, Aug 19 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 24 2016
STATUS
approved