A317876 - OEIS

%I #15 Apr 30 2019 21:50:27

%S 1,1,2,4,10,25,67,184,519,1489,4342,12812,38207,114934,348397,1063050,

%T 3262588,10064645,31190985,97061431,303165207,950115502,2986817742,

%U 9415920424,29760442192,94286758293,299377379027,952521579944,3036380284111,9696325863803

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

%C A free pure symmetric identity multifunction (with empty expressions allowed) (FOI) is either (case 1) the leaf symbol "o", or (case 2) a possibly empty expression of the form h[g_1, ..., g_k] where h is an FOI, each of the g_i for i = 1, ..., k >= 0 is an FOI, and for i < j we have g_i < g_j under a canonical total ordering such as the Mathematica ordering of expressions. The number of positions in an FOI is the number of brackets [...] plus the number of o's.

%C Also the number of free orderless identity Mathematica expressions with one atom and n positions.

%H Andrew Howroyd, <a href="/A317876/b317876.txt">Table of n, a(n) for n = 1..200</a>

%F From _Ilya Gutkovskiy_, Apr 30 2019: (Start)

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

%F G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x * (1 + (Sum_{n>=1} a(n)*x^n) * Product_{n>=1} (1 + x^n)^a(n)). (End)

%e The a(5) = 10 FOIs:

%e   o[o[o]]

%e   o[o][o]

%e   o[o[][]]

%e   o[o,o[]]

%e   o[][o[]]

%e   o[][][o]

%e   o[o[]][]

%e   o[][o][]

%e   o[o][][]

%e   o[][][][]

%t allIdExpr[n_]:=If[n==1,{"o"},Join@@Cases[Table[PR[k,n-k-1],{k,n-1}],PR[h_,g_]:>Join@@Table[Apply@@@Tuples[{allIdExpr[h],Select[Union[Sort/@Tuples[allIdExpr/@p]],UnsameQ@@#&]}],{p,IntegerPartitions[g]}]]];

%t Table[Length[allIdExpr[n]],{n,12}]

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

%o seq(n)={my(v=[1]); for(n=2, n, my(t=WeighT(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 Cf. A000081, A004111, A052893, A053492, A277996, A280000, A317652, A317653, A317654, A317875.

%Y Cf. A317877, A317878, A317879, A317880, A317881.

%K nonn

%O 1,3

%A _Gus Wiseman_, Aug 09 2018

%E Terms a(16) and beyond from _Andrew Howroyd_, Aug 19 2018