cp's OEIS Frontend

A277996 Number of free pure symmetric multifunctions (with empty expressions allowed) with one atom and n positions.

%I A277996 #33 Apr 30 2019 21:50:00
%S A277996 1,1,2,5,13,36,102,299,892,2713,8364,26108,82310,261804,838961,
%T A277996 2706336,8780725,28636157,93818641,308641277,1019140129,3376604826,
%U A277996 11221805968,37399728251,124967677989,418564867751,1405030366113,4726036692421,15927027834163,53770343259613
%N A277996 Number of free pure symmetric multifunctions (with empty expressions allowed) with one atom and n positions.
%C A277996 Also the number of distinct orderless Mathematica expressions with one atom and n positions.
%H A277996 Andrew Howroyd, <a href="/A277996/b277996.txt">Table of n, a(n) for n = 1..200</a>
%H A277996 Mathematica Reference, <a href="http://reference.wolfram.com/mathematica/ref/Orderless.html">Orderless</a>.
%F A277996 From _Ilya Gutkovskiy_, Apr 30 2019: (Start)
%F A277996 G.f. A(x) satisfies: A(x) = x * (1 + A(x) * exp(Sum_{k>=1} A(x^k)/k)).
%F A277996 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)
%e A277996 The a(5)=13 Mathematica expressions are:
%e A277996 x[x,x,x]
%e A277996 x[x,x][]   x[x][x]   x[][x,x]  x[x,x[]]  x[x[x]]
%e A277996 x[x][][]   x[][x][]  x[][][x]  x[x[]][]  x[][x[]]  x[x[][]]
%e A277996 x[][][][]
%t A277996 multing[t_,n_]:=Array[(t+#-1)/#&,n,1,Times];
%t A277996 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}]];
%t A277996 Array[a,30]
%o A277996 (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
%o A277996 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
%Y A277996 Cf. A000108, A001003, A005043, A052893, A279944, A280000, A317658.
%K A277996 nonn
%O A277996 1,3
%A A277996 _Gus Wiseman_, Dec 24 2016