A055314 - OEIS

%I #42 Aug 14 2024 01:53:53

%S 1,3,0,12,4,0,60,60,5,0,360,720,210,6,0,2520,8400,5250,630,7,0,20160,

%T 100800,109200,30240,1736,8,0,181440,1270080,2116800,1058400,151704,

%U 4536,9,0,1814400,16934400,40219200,31752000,8573040,695520,11430,10,0

%N Triangle T(n,k) read by rows: number of labeled trees with n nodes and k leaves, n >= 2, 2 <= k <= n.

%D 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

%D 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 D. Stanton and D. White, Constructive Combinatorics, Springer, 1986; see p. 67.

%H G. C. Greubel, <a href="/A055314/b055314.txt">Table of n, a(n) for the first 50 rows, flattened</a>

%H A. M. Hamel, <a href="http://dx.doi.org/10.1007/s000260300004">Priority queue sorting and labeled trees</a>, Annals Combin., 7 (2003), 49-54.

%H F. Harary, A. Mowshowitz and J. Riordan, <a href="https://doi.org/10.1016/S0021-9800(69)80106-7">Labeled trees with unlabeled end-points</a>, J. Combin. Theory, 6 (1969), 60-64. - From _N. J. A. Sloane_, Jun 07 2012

%H Vites Longani, <a href="http://dx.doi.org/10.1016/j.camwa.2008.07.011">A formula for the number of labelled trees</a>, Comp. Math. Appl. 56 (2008) 2786-2788.

%H John Riordan, <a href="https://doi.org/10.1090/S0002-9904-1966-11442-8">The enumeration of labeled trees by degrees</a>, Bull. Amer. Math. Soc. 72 1966 110--112. MR0186583 (32 #4042). - From _N. J. A. Sloane_, Jun 07 2012

%H <a href="/index/Tra#trees">Index entries for sequences related to trees</a>

%F 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.

%F 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).

%F T(n, k) = (n!/k!)*Stirling2(n-2, n-k). - _Vladeta Jovovic_, Jan 28 2004

%e Triangle T(n,k) begins:

%e      1;

%e      3,    0;

%e     12,    4,    0;

%e     60,   60,    5,   0;

%e    360,  720,  210,   6, 0;

%e   2520, 8400, 5250, 630, 7, 0;

%e   ...

%p 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);

%p # 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

%p with(combinat);

%p R:=proc(n,k)

%p if n=1 then if k=1 then RETURN(1) else RETURN(0); fi

%p     elif (n=2 and k=2) then RETURN(1)

%p     elif (n=2 and k>2) then RETURN(0)

%p     else stirling2(n-2,n-k)*n!/k!;

%p     fi;

%p end;

%t 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 *)

%t Table[(n!/k!)*StirlingS2[n - 2, n - k], {n, 2, 7}, {k, 2, n}]//Flatten (* _G. C. Greubel_, May 17 2017 *)

%o (Maxima) A055314(n,k) := block(

%o         n!/k!*stirling2(n-2,n-k)

%o )$

%o for n : 2 thru 10 do

%o         for k : 2 thru n do

%o                 print(A055314(n,k)," ") ; /* _R. J. Mathar_, Mar 06 2012 */

%o (PARI) for(n=2,20, for(k=2,n, print1((n!/k!)*stirling(n-2, n-k, 2), ", "))) \\ _G. C. Greubel_, May 17 2017

%Y Cf. A000272.

%Y Row sums give A000272. Columns 2 through 12: A001710, A055315-A055324.

%K nonn,tabl

%O 2,2

%A _Christian G. Bower_, May 11 2000