cp's OEIS Frontend

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

%I A280000 #8 Aug 19 2018 16:46:12
%S A280000 1,0,1,1,3,5,12,25,57,128,296,688,1618,3839,9170,22065,53370,129807,
%T A280000 317080,777887,1915247,4731932,11726476,29143123,72614115,181363151,
%U A280000 453975928,1138697689,2861607677,7204169689
%N A280000 Number of free pure symmetric multifunctions in one symbol with n positions.
%C A280000 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 A280000 Andrew Howroyd, <a href="/A280000/b280000.txt">Table of n, a(n) for n = 1..200</a>
%e A280000 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 A280000 1 x            1
%e A280000 3 x[x]         2
%e A280000 4 x[x,x]       8
%e A280000 5 x[x][x]      3
%e A280000 5 x[x[x]]      4
%e A280000 5 x[x,x,x]     128
%e A280000 6 x[x,x][x]    12
%e A280000 6 x[x][x,x]    27
%e A280000 6 x[x,x[x]]    32
%e A280000 6 x[x,x,x,x]   32768
%e A280000 6 x[x[x,x]]    262144
%e A280000 7 x[x][x][x]   5
%e A280000 7 x[x[x]][x]   6
%e A280000 7 x[x][x[x]]   9
%e A280000 7 x[x[x][x]]   16
%e A280000 7 x[x[x[x]]]   64
%e A280000 7 x[x,x,x][x]  145
%e A280000 7 x[x,x][x,x]  1728
%e A280000 7 x[x,x,x[x]]  2048
%e A280000 7 x[x][x,x,x]  2187
%e A280000 7 x[x,x,x,x,x] 2147483648
%e A280000 7 x[x,x[x,x]]  137438953472
%e A280000 7 x[x[x,x,x]]  1378913...3030144
%t A280000 multing[t_,n_]:=Array[(t+#-1)/#&,n,1,Times];
%t A280000 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 A280000 Array[a,15]
%o A280000 (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
%o A280000 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 A280000 Cf. A005043 (non-symmetric case), A279944.
%K A280000 nonn
%O A280000 1,5
%A A280000 _Gus Wiseman_, Dec 24 2016