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A002861 Number of connected functions (or mapping patterns) on n unlabeled points, or number of rings and branches with n edges.
(Formerly M1182 N0455)
16

%I M1182 N0455

%S 1,2,4,9,20,51,125,329,862,2311,6217,16949,46350,127714,353272,981753,

%T 2737539,7659789,21492286,60466130,170510030,481867683,1364424829,

%U 3870373826,10996890237,31293083540,89173833915,254445242754,726907585652,2079012341822

%N Number of connected functions (or mapping patterns) on n unlabeled points, or number of rings and branches with n edges.

%C A000081 + A027852 + A029852 + A029853 + A029868 + ... - _Geoffrey Critzer_, Oct 12 2012

%D S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 5.6.6.

%D R. A. Fisher, Contributions to Mathematical Statistics, Wiley, 1950, 41.399.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Alois P. Heinz, <a href="/A002861/b002861.txt">Table of n, a(n) for n = 1..1000</a> (first 500 terms from C. G. Bower)

%H C. G. Bower, <a href="/transforms2.html">Transforms (2)</a>

%H P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/books.html">Analytic Combinatorics</a>, 2009; see page 480

%H R. K. Guy, <a href="/A000081/a000081.pdf">Letter to N. J. A. Sloane, 1988-04-12</a> (annotated scanned copy)

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=118">Encyclopedia of Combinatorial Structures 118</a>

%F CIK transform of A000081.

%p spec2861 := [B, {A=Prod(Z,Set(A)), B=Cycle(A)}, unlabeled]; [seq(combstruct[count](spec2861,size=n), n=1..27)];

%t Needs["Combinatorica`"];

%t nn=30;s[n_,k_]:=s[n,k]=a[n+1-k]+If[n<2 k,0,s[n-k,k]];a[1]=1;a[n_]:=a[n]=Sum[a[i] s[n-1,i] i,{i,1,n-1}]/(n-1);rt=Table[a[i],{i,1,nn}];Apply[Plus,Table[Take[CoefficientList[CycleIndex[CyclicGroup[n],s]/.Table[s[j]->Table[Sum[rt[[i]] x^(k*i),{i,1,nn}],{k,1,nn}][[j]],{j,1,nn}],x],nn],{n,1,30}]] (* _Geoffrey Critzer_, Oct 12 2012, after code given by _Robert A. Russell_ in A000081 *)

%o (PARI)

%o N=66; A=vector(N+1, j, 1);

%o for (n=1, N, A[n+1] = 1/n * sum(k=1, n, sumdiv(k, d, d * A[d]) * A[n-k+1] ) );

%o A000081=concat([0], A);

%o H(t)=subst(Ser(A000081, 't), 't, t);

%o x='x+O('x^N);

%o L=sum(j=1,N, eulerphi(j)/j * log(1/(1-H(x^j))));

%o Vec(L)

%o \\ _Joerg Arndt_, Jul 10 2014

%Y Cf. A000081, A001372.

%K nonn,nice

%O 1,2

%A _N. J. A. Sloane_

%E More terms from _Philippe Flajolet_ and _Paul Zimmermann_, Mar 15 1996

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Last modified April 18 22:08 EDT 2019. Contains 322237 sequences. (Running on oeis4.)