BBC Russian

A002861

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

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)` `A. L. Agore, A. Chirvasitu, and G. Militaru, The set-theoretic Yang-Baxter equation, Kimura semigroups and functional graphs, arXiv:2303.06700 [math.QA], 2023.` `C. G. Bower, Transforms (2)` `Oscar Defrain, Antonio E. Porreca and Ekaterina Timofeeva, Polynomial-delay generation of functional digraphs up to isomorphism, Disc. Appl. Math., vol 357 (2024), pp. 24-33.` `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-François 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	`Row sums of A339428.` `Cf. A000081, A001372.` `Cf. A027852, A029852, A029853, A029868.` `Sequence in context: A134955 A171887 A027881 * A363203 A032200 A130969` `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`