A000243 Number of trees with n nodes, 2 of which are labeled. (Formerly M2803 N1128)

1, 3, 9, 26, 75, 214, 612, 1747, 4995, 14294, 40967, 117560, 337830, 972027, 2800210, 8075889, 23315775, 67380458, 194901273, 564239262, 1634763697, 4739866803, 13752309730, 39926751310, 115988095896, 337138003197

Offset

2,2

References

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 138.

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

T. D. Noe, Table of n, a(n) for n=2..200

R. J. Mathar, Topologically distinct sets of non-intersecting circles in the plane, arXiv:1603.00077 [math.CO] (2016), Table 5.

Index entries for sequences related to rooted trees

Index entries for sequences related to trees

Formula

a(n) = A000107(n) - A000081(n). - Christian G. Bower, Nov 15 1999

G.f.: A(x) = B(x)^2/(1-B(x)), where B(x) is g.f. for rooted trees with n nodes, cf. A000081. - Vladeta Jovovic, Oct 19 2001

a(n) = A000106(n) + A304068(n). - Brendan McKay, May 05 2018

Maple

b:= proc(n) option remember; if n<=1 then n else add(k*b(k)* s(n-1, k), k=1..n-1)/(n-1) fi end: s:= proc(n, k) option remember; add(b(n+1-j*k), j=1..iquo(n, k)) end: B:= proc(n) option remember; add(b(k)*x^k, k=1..n) end: a:= n-> coeff(series(B(n-1)^2/(1-B(n-1)), x=0, n+1), x, n): seq(a(n), n=2..27); # Alois P. Heinz, Aug 21 2008

Mathematica

b[n_] := b[n] = If[ n <= 1 , n, Sum[k*b[k]*s[n - 1, k], {k, 1, n - 1}]/(n - 1) ]; s[n_, k_] := s[n, k] = Sum[ b[n + 1 - j*k], {j, 1, Quotient[n, k]}]; B[n_] := B[n] = Sum[ b[k]*x^k, {k, 1, n}]; a[n_] := Coefficient[ Series[ B[n - 1]^2/(1 - B[n - 1]), {x, 0, n + 1}], x, n]; Table[ a[n], {n, 2, 27}] (* Jean-François Alcover, Jan 25 2012, translated from Maple *)

Crossrefs

Column k=2 of A034799.

Sequence in context: A171277 A289806 A303976 * A076264 A018919 A123941

Adjacent sequences: A000240 A000241 A000242 * A000244 A000245 A000246

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

More terms and new description from Christian G. Bower, Nov 15 1999

Status

approved