A280000 Number of free pure symmetric multifunctions in one symbol with n positions.

1, 0, 1, 1, 3, 5, 12, 25, 57, 128, 296, 688, 1618, 3839, 9170, 22065, 53370, 129807, 317080, 777887, 1915247, 4731932, 11726476, 29143123, 72614115, 181363151, 453975928, 1138697689, 2861607677, 7204169689

Offset: 1

Author

Gus Wiseman, Dec 24 2016

Keywords

• nonn

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.

Examples

			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
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..200

Crossrefs

Cf. A005043 (non-symmetric case), A279944.

Programs

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]

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