A280000 - OEIS

%I #8 Aug 19 2018 16:46:12

%S 1,0,1,1,3,5,12,25,57,128,296,688,1618,3839,9170,22065,53370,129807,

%T 317080,777887,1915247,4731932,11726476,29143123,72614115,181363151,

%U 453975928,1138697689,2861607677,7204169689

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

%C 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.

%H Andrew Howroyd, <a href="/A280000/b280000.txt">Table of n, a(n) for n = 1..200</a>

%e 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:

%e 1 x            1

%e 3 x[x]         2

%e 4 x[x,x]       8

%e 5 x[x][x]      3

%e 5 x[x[x]]      4

%e 5 x[x,x,x]     128

%e 6 x[x,x][x]    12

%e 6 x[x][x,x]    27

%e 6 x[x,x[x]]    32

%e 6 x[x,x,x,x]   32768

%e 6 x[x[x,x]]    262144

%e 7 x[x][x][x]   5

%e 7 x[x[x]][x]   6

%e 7 x[x][x[x]]   9

%e 7 x[x[x][x]]   16

%e 7 x[x[x[x]]]   64

%e 7 x[x,x,x][x]  145

%e 7 x[x,x][x,x]  1728

%e 7 x[x,x,x[x]]  2048

%e 7 x[x][x,x,x]  2187

%e 7 x[x,x,x,x,x] 2147483648

%e 7 x[x,x[x,x]]  137438953472

%e 7 x[x[x,x,x]]  1378913...3030144

%t multing[t_,n_]:=Array[(t+#-1)/#&,n,1,Times];

%t 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}]];

%t Array[a,15]

%o (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}

%o 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

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

%K nonn

%O 1,5

%A _Gus Wiseman_, Dec 24 2016