2,3

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).

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.

# 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)];

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 *)

Cf. A151880.

Sequence in context: A317022 A143917 A009326 * A247499 A124373 A010787

Adjacent sequences: A001255 A001256 A001257 * A001259 A001260 A001261

nonn,nice

N. J. A. Sloane. More terms from N. J. A. Sloane, Jun 07 2012

approved