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

1, 1, 2, 5, 13, 36, 102, 299, 892, 2713, 8364, 26108, 82310, 261804, 838961, 2706336, 8780725, 28636157, 93818641, 308641277, 1019140129, 3376604826, 11221805968, 37399728251, 124967677989, 418564867751, 1405030366113, 4726036692421, 15927027834163, 53770343259613

Offset: 1

Author

Gus Wiseman, Dec 24 2016

Keywords

• nonn

Comments

Also the number of distinct orderless Mathematica expressions with one atom and n positions.

Examples

			The a(5)=13 Mathematica expressions are:
x[x,x,x]
x[x,x][]   x[x][x]   x[][x,x]  x[x,x[]]  x[x[x]]
x[x][][]   x[][x][]  x[][][x]  x[x[]][]  x[][x[]]  x[x[][]]
x[][][][]
		

Links

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

Crossrefs

Cf. A000108, A001003, A005043, A052893, A279944, A280000, A317658.

Programs

Mathematica

multing[t_,n_]:=Array[(t+#-1)/#&,n,1,Times];
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}]];
Array[a,30]

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, v[n-1] + sum(k=1, n-2, v[k]*t[n-k-1]))); v} \\ Andrew Howroyd, Aug 19 2018

Formula

From Ilya Gutkovskiy, Apr 30 2019: (Start)

G.f. A(x) satisfies: A(x) = x * (1 + A(x) * exp(Sum_{k>=1} A(x^k)/k)).

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)