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A055314
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Triangle T(n,k) read by rows: number of labeled trees with n nodes and k leaves, n >= 2, 2 <= k <= n.
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23
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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
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OFFSET
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2,2
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REFERENCES
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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.
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LINKS
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FORMULA
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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).
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EXAMPLE
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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;
...
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MAPLE
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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
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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 *)
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PROG
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n!/k!*stirling2(n-2, n-k)
)$
for n : 2 thru 10 do
for k : 2 thru n do
(PARI) for(n=2, 20, for(k=2, n, print1((n!/k!)*stirling(n-2, n-k, 2), ", "))) \\ G. C. Greubel, May 17 2017
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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