OFFSET
1,5
COMMENTS
A free pure symmetric multifunction (PSM) in one symbol x is either (case 1) the symbol x, or (case 2) an expression of the form h[g_1,...,g_k] where h is a PSM in x, each of the g_i for i=1..(k>0) is a PSM in x, and for i < j we have g_i <= g_j under a canonical total ordering such as the Mathematica ordering. The number of positions in a PSM is the number of brackets [...] plus the number of x's.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..200
EXAMPLE
Sequence of free pure symmetric multifunctions (second column) together with their numbers of positions (first column) and j-numbers (third column, see A279944 for details) begins:
1 x 1
3 x[x] 2
4 x[x,x] 8
5 x[x][x] 3
5 x[x[x]] 4
5 x[x,x,x] 128
6 x[x,x][x] 12
6 x[x][x,x] 27
6 x[x,x[x]] 32
6 x[x,x,x,x] 32768
6 x[x[x,x]] 262144
7 x[x][x][x] 5
7 x[x[x]][x] 6
7 x[x][x[x]] 9
7 x[x[x][x]] 16
7 x[x[x[x]]] 64
7 x[x,x,x][x] 145
7 x[x,x][x,x] 1728
7 x[x,x,x[x]] 2048
7 x[x][x,x,x] 2187
7 x[x,x,x,x,x] 2147483648
7 x[x,x[x,x]] 137438953472
7 x[x[x,x,x]] 1378913...3030144
MATHEMATICA
multing[t_, n_]:=Array[(t+#-1)/#&, n, 1, Times];
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-2}]];
Array[a, 15]
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, 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