A055314 Triangle T(n,k) read by rows: number of labeled trees with n nodes and k leaves, n >= 2, 2 <= k <= n.
1, 3, 0, 12, 4, 0, 60, 60, 5, 0, 360, 720, 210, 6, 0, 2520, 8400, 5250, 630, 7, 0, 20160, 100800, 109200, 30240, 1736, 8, 0, 181440, 1270080, 2116800, 1058400, 151704, 4536, 9, 0, 1814400, 16934400, 40219200, 31752000, 8573040, 695520, 11430, 10, 0
Offset: 2
Examples
Triangle T(n,k) begins:
1;
3, 0;
12, 4, 0;
60, 60, 5, 0;
360, 720, 210, 6, 0;
2520, 8400, 5250, 630, 7, 0;
...
References
- Moon, J. W. Various proofs of Cayley's formula for counting trees. 1967 A seminar on Graph Theory pp. 70--78 Holt, Rinehart and Winston, New York; MR0214515 (35 #5365). - From N. J. A. Sloane, Jun 07 2012
- Renyi, Alfred. Some remarks on the theory of trees. Magyar Tud. Akad. Mat. Kutató Int. Közl. 4 1959 73--85. MR0115938 (22 #6735). - From N. J. A. Sloane, Jun 07 2012
- D. Stanton and D. White, Constructive Combinatorics, Springer, 1986; see p. 67.
Links
- G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
- A. M. Hamel, Priority queue sorting and labeled trees, Annals Combin., 7 (2003), 49-54.
- F. Harary, A. Mowshowitz and J. Riordan, Labeled trees with unlabeled end-points, J. Combin. Theory, 6 (1969), 60-64. - From _N. J. A. Sloane_, Jun 07 2012
- Vites Longani, A formula for the number of labelled trees, Comp. Math. Appl. 56 (2008) 2786-2788.
- John Riordan, The enumeration of labeled trees by degrees, Bull. Amer. Math. Soc. 72 1966 110--112. MR0186583 (32 #4042). - From _N. J. A. Sloane_, Jun 07 2012
- Index entries for sequences related to trees
Programs
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Maple
T := (n,k) -> binomial(n+1,k)*add( binomial(n+1-k,i)*(-1)^(n+1-k-i)*i^(n-1), i=0..n+1-k); # The following version gives the triangle for any n>=1, k>=1, based on the Harary et al. (1969) paper - N. J. A. Sloane, Jun 07 2012 with(combinat); R:=proc(n,k) 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;
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Mathematica
Table[Table[Binomial[n,k] Sum[(-1)^j Binomial[n-k,j] (n-k-j)^(n-2),{j,0,n-k}],{k,2,n-1}],{n,2,10}]//Grid (* Geoffrey Critzer, Nov 12 2011 *) Table[(n!/k!)*StirlingS2[n - 2, n - k], {n, 2, 7}, {k, 2, n}]//Flatten (* G. C. Greubel, May 17 2017 *) -
Maxima
A055314(n,k) := block( n!/k!*stirling2(n-2,n-k) )$ for n : 2 thru 10 do for k : 2 thru n do print(A055314(n,k)," ") ; /* R. J. Mathar, Mar 06 2012 */
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PARI
for(n=2,20, for(k=2,n, print1((n!/k!)*stirling(n-2, n-k, 2), ", "))) \\ G. C. Greubel, May 17 2017
Formula
E.g.f.: A(x, y)=(1-x+x*y)*B(x, y)-B(x, y)^2/2. B(x, y): e.g.f. of A055302.
T(n, k) = binomial(n+1, k)*Sum( binomial(n+1-k, i)*(-1)^(n+1-k-i)*i^(n-1), i=0..n+1-k).
T(n, k) = (n!/k!)*Stirling2(n-2, n-k). - Vladeta Jovovic, Jan 28 2004