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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
1, 2, 4, 9, 20, 51, 125, 329, 862, 2311, 6217, 16949, 46350, 127714, 353272, 981753, 2737539, 7659789, 21492286, 60466130, 170510030, 481867683, 1364424829, 3870373826, 10996890237, 31293083540, 89173833915, 254445242754, 726907585652, 2079012341822 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

A000081 + A027852 + A029852 + A029853 + A029868 + ... - Geoffrey Critzer, Oct 12 2012

REFERENCES

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

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

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

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

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..1000 (first 500 terms from C. G. Bower)

C. G. Bower, Transforms (2)

Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, 2009; see page 480

R. K. Guy, Letter to N. J. A. Sloane, 1988-04-12 (annotated scanned copy)

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 118

FORMULA

CIK transform of A000081.

MAPLE

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

MATHEMATICA

Needs["Combinatorica`"];

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, nn}], {k, 1, nn}][[j]], {j, nn}], x], nn], {n, 30}]]  (* Geoffrey Critzer, Oct 12 2012, after code given by Robert A. Russell in A000081 *)

M = 66; A = Table[1, {M + 1}]; For[n = 1, n <= M, n++, A[[n + 1]] = 1/n * Sum[Sum[d * A[[d]], {d, Divisors[k]}] * A[[n - k + 1]], {k, n}]]; A81 = {0} ~ Join ~ A; H[t_] = A81.t^Range[0, Length[A81] - 1]; L = Sum[EulerPhi[j]/j * Log[1/(1 - H[x^j])], {j, M}] + O[x]^M; CoefficientList[L, x] // Rest (* Jean-Fran├žois Alcover, Dec 28 2019, after Joerg Arndt *)

PROG

(PARI)

N=66;  A=vector(N+1, j, 1);

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

A000081=concat([0], A);

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

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

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

Vec(L)

\\ Joerg Arndt, Jul 10 2014

CROSSREFS

Cf. A000081, A001372.

Sequence in context: A134955 A171887 A027881 * A032200 A130969 A264293

Adjacent sequences:  A002858 A002859 A002860 * A002862 A002863 A002864

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Philippe Flajolet and Paul Zimmermann, Mar 15 1996

STATUS

approved

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Last modified October 28 06:26 EDT 2020. Contains 338048 sequences. (Running on oeis4.)