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 A027852 Number of connected functions on n points with a loop of length 2. 19
 0, 1, 1, 3, 6, 16, 37, 96, 239, 622, 1607, 4235, 11185, 29862, 80070, 216176, 586218, 1597578, 4370721, 12003882, 33077327, 91433267, 253454781, 704429853, 1962537755, 5479855546, 15332668869, 42983656210, 120716987723, 339596063606, 956840683968 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS Number of unordered pairs of rooted trees with a total of n nodes. Equivalently, the number of rooted trees on n+1 nodes where the root has degree 2. Number of trees on n nodes rooted at an edge. - Washington Bomfim, Jul 06 2012 Guy (1988) calls these tadpole graphs. - N. J. A. Sloane, Nov 04 2014 Number of unicyclic graphs of n nodes with a cycle length of two (in other words, a double edge). - Washington Bomfim, Dec 02 2020 LINKS Alois P. Heinz, Table of n, a(n) for n = 1..2136 Washington Bomfim, Illustration of initial terms R. K. Guy, Letter to N. J. A. Sloane, 1988-04-12 (annotated scanned copy) Includes illustrations for n <= 6. R. J. Mathar, Topologically distinct sets of non-intersecting circles in the plane, arXiv:1603.00077 [math.CO] (2016), Eq. (75). FORMULA G.f.: A(x) = (B(x)^2 + B(x^2))/2 where B(x) is g.f. of A000081. a(n) = Sum_{k=1..(n-1)/2}( f(k)*f(n-k) ) + [n mod 2 = 0] * ( f(n/2)^2+f(n/2) ) /2, where f(n) = A000081(n). - Washington Bomfim, Jul 06 2012 and Dec 01 2020 a(n) ~ c * d^n / n^(3/2), where d = A051491 = 2.9557652856519949747148..., c = A187770 = 0.43992401257102530404090339... . - Vaclav Kotesovec, Sep 12 2014 2*a(n) = A000106(n) + A000081(n/2), where A(.)=0 if the argument is non-integer. - R. J. Mathar, Jun 04 2020 MAPLE with(numtheory): b:= proc(n) option remember; local d, j; `if`(n<=1, n, (add(add(d*b(d), d=divisors(j)) *b(n-j), j=1..n-1))/ (n-1)) end: a:= n-> (add(b(i) *b(n-i), i=0..n) +`if`(irem(n, 2)=0, b(n/2), 0))/2: seq(a(n), n=1..50); # Alois P. Heinz, Aug 22 2008, revised Oct 07 2011 # second, re-usable version A027852 := proc(N::integer) local dh, Nprime; dh := 0 ; for Nprime from 0 to N do dh := dh+A000081(Nprime)*A000081(N-Nprime) ; end do: if type(N, 'even') then dh := dh+A000081(N/2) ; end if; dh/2 ; end proc: # R. J. Mathar, Mar 06 2017 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; 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}]; Take[CoefficientList[CycleIndex[DihedralGroup, s] /. Table[s[j] -> Table[Sum[rt[[i]] x^(k*i), {i, 1, nn}], {k, 1, nn}][[j]], {j, 1, nn}], x], {2, nn}] (* Geoffrey Critzer, Oct 12 2012, after code given by Robert A. Russell in A000081 *) b[n_] := b[n] = If[n <= 1, n, (Sum[Sum[d b[d], {d, Divisors[j]}] b[n-j], {j, 1, n-1}])/(n-1)]; a[n_] := (Sum[b[i] b[n-i], {i, 0, n}] + If[Mod[n, 2] == 0, b[n/2], 0])/2; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Oct 30 2018, after Alois P. Heinz *) PROG (PARI) seq(max_n)= { my(V = f = vector(max_n), i=1, s); f=1; for(j=1, max_n - 1, f[j+1] = 1/j * sum(k=1, j, sumdiv(k, d, d * f[d]) * f[j-k+1])); for(n = 1, max_n, s = sum(k = 1, (n-1)/2, ( f[k] * f[n-k] )); if(n % 2 == 1, V[i] = s, V[i] = s + (f[n/2]^2 + f[n/2])/2); i++); V }; \\ Washington Bomfim, Jul 06 2012 and Dec 01 2020 CROSSREFS Column 2 of A033185 (forests of rooted trees), A217781 (unicyclic graphs), A339303 (unoriented linear forests) and A339428 (connected functions). Cf. A000081, A000106, A000226, A000631, A001372, A002861. Sequence in context: A072824 A360229 A089406 * A203068 A321229 A114410 Adjacent sequences: A027849 A027850 A027851 * A027853 A027854 A027855 KEYWORD nonn AUTHOR Christian G. Bower, Dec 14 1997 EXTENSIONS Edited by Christian G. Bower, Feb 12 2002 STATUS approved

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Last modified March 22 17:52 EDT 2023. Contains 361432 sequences. (Running on oeis4.)