OFFSET
2,3
REFERENCES
J.W. Moon, Counting Labelled Trees, Issue 1 of Canadian mathematical monographs, Canadian Mathematical Congress, 1970, Sec. 3.9.
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
N. J. A. Sloane, Table of n, a(n) for n = 2..100
F. Harary, A. Mowshowitz and J. Riordan, Labeled trees with unlabeled end-points, J. Combin. Theory, 6 (1969), 60-64.
MAPLE
# This gives the sequence but without the initial 1:
with(combinat);
R:=proc(n, k) # this gives A055314
if n=1 then if k=1 then RETURN(1) else RETURN(0); fi
elif (n=2 and k=2) then RETURN(1)
elif (n=2 and k>2) then RETURN(0)
else stirling2(n-2, n-k)*n!/k!;
fi;
end;
Rstar:=proc(n, k) # this gives A213262
if k=2 then
if n <=4 then RETURN(1); else RETURN((n-2)!/2); fi;
else
if k <= n-2 then add(binomial(n-i-1, k-i)*R(n-k, i), i=2..n-1);
elif k=n-1 then 1;
else 0;
fi;
fi;
end;
[seq(add(Rstar(n, k), k=2..n-1), n=3..20)];
MATHEMATICA
r[n_, k_] := Which[n == 1, If[k == 1, Return[1], Return[0]], n == 2 && k == 2, Return[1], n == 2 && k > 2, Return[0], n > k > 0, StirlingS2[n-2, n-k]*n!/k!, True, 0]; rstar[n_, k_] := Which[k == 2, If[n <= 4, Return[1], Return[(n-2)!/2]], k <= n-2, Sum[Binomial[n-i-1, k-i]*r[n-k, i], {i, 2, n-1}], k == n-1, 1, True, 0]; Join[{1}, Table[Sum[rstar[n, k], {k, 2, n-1}], {n, 3, 26}]] (* Jean-François Alcover, Oct 08 2012, translated from Maple *)
tStar[2] = 1;
tStar[n_] :=
Sum[(-1)^j Binomial[n - k, j] Binomial[n - 1 - j,
k] (n - k - j)^(n - k - 2), {k, 2, n - 1}, {j, 0, n - k - 1}];
Table[tStar[n], {n, 2, 20}] (* David Callan, Jul 18 2014, after Moon reference *)
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
N. J. A. Sloane. More terms from N. J. A. Sloane, Jun 07 2012
STATUS
approved