cp's OEIS Frontend

A317880 Number of series-reduced free pure symmetric identity multifunctions (with empty expressions allowed) with one atom and n positions.

%I A317880 #12 Sep 11 2018 21:17:26
%S A317880 1,1,1,1,2,4,8,16,33,70,152,333,735,1635,3668,8285,18823,42970,98535,
%T A317880 226870,524290,1215641,2827203,6593432,15416197,36129894,84860282,
%U A317880 199719932,470930802,1112388190,2631903295,6236669381,14800078408,35169529363,83680908692
%N A317880 Number of series-reduced free pure symmetric identity multifunctions (with empty expressions allowed) with one atom and n positions.
%C A317880 A series-reduced free pure symmetric identity multifunction (with empty expressions allowed) (SROI) 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 SROI, k is an integer greater than or equal to 0 but not equal to 1, each of the g_i for i = 1, ..., k is an SROI, 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 SROI is the number of brackets [...] plus the number of o's.
%C A317880 Also the number of series-reduced orderless identity Mathematica expressions with one atom and n positions.
%H A317880 Andrew Howroyd, <a href="/A317880/b317880.txt">Table of n, a(n) for n = 1..200</a>
%e A317880 The a(7) = 8 SROIs:
%e A317880   o[o,o[][][]]
%e A317880   o[o[],o[][]]
%e A317880   o[][o,o[][]]
%e A317880   o[][][o,o[]]
%e A317880   o[o,o[][]][]
%e A317880   o[][o,o[]][]
%e A317880   o[o,o[]][][]
%e A317880   o[][][][][][]
%t A317880 allIdExprSR[n_]:=If[n==1,{"o"},Join@@Cases[Table[PR[k,n-k-1],{k,n-1}],PR[h_,g_]:>Join@@Table[Apply@@@Tuples[{allIdExprSR[h],Select[Union[Sort/@Tuples[allIdExprSR/@p]],UnsameQ@@#&]}],{p,If[g==0,{{}},Rest[IntegerPartitions[g]]]}]]];
%t A317880 Table[Length[allIdExprSR[n]],{n,12}]
%o A317880 (PARI) WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
%o A317880 seq(n)={my(v=[1]); for(n=2, n, my(t=WeighT(v)-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 A317880 Cf. A001003, A001678, A052893, A053492, A067824, A167865, A277996, A280000, A317875.
%Y A317880 Cf. A317876, A317877, A317878, A317879, A317880, A317881.
%K A317880 nonn
%O A317880 1,5
%A A317880 _Gus Wiseman_, Aug 09 2018
%E A317880 Terms a(13) and beyond from _Andrew Howroyd_, Aug 19 2018