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 A055302 Triangle of number of labeled rooted trees with n nodes and k leaves, n >= 1, 1 <= k <= n. 23
 1, 2, 0, 6, 3, 0, 24, 36, 4, 0, 120, 360, 140, 5, 0, 720, 3600, 3000, 450, 6, 0, 5040, 37800, 54600, 18900, 1302, 7, 0, 40320, 423360, 940800, 588000, 101136, 3528, 8, 0, 362880, 5080320, 16087680, 15876000, 5143824, 486864, 9144, 9, 0, 3628800 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Beginning with the second row, dividing each row by n gives the mirror of row n-1 of A141618. Under the exponential transform, the mirror of A141618 is generated, relating the number of connected graphs here to the number of disconnected graphs associated with A141618 (cf. A127671 and A036040). - Tom Copeland, Oct 25 2014 REFERENCES Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 313. LINKS Alois P. Heinz, Rows n = 1..141, flattened N. J. A. Sloane, Transforms FORMULA E.g.f. (relative to x) satisfies: A(x,y) = xy + x*exp(A(x,y)) - x. Divides by n and shifts up under exponential transform. T(n,k) = (n!/k!)*Stirling2(n-1, n-k). - Vladeta Jovovic, Jan 28 2004 T(n,k) = A055314(n,k)*(n-k) + A055314(n,k+1)*(k+1). The first term is the number of such trees with root degree > 1 while the second term is the number of such trees with root degree = 1. This simplifies to the above formula by Vladeta Jovovic. - Geoffrey Critzer, Dec 01 2012 E.g.f.: G(x,t) = log[1 + t * N(x*t,1/t)], where N(x,t) is the e.g.f. of A141618. Also, G(x*t,1/t)= log[1 + N(x,t)/t] is the comp. inverse in x of x / [1 + t * (e^x - 1)]. - Tom Copeland, Oct 26 2014 EXAMPLE Triangle begins      1,      2,     0;      6,     3,     0;     24,    36,     4,     0;    120,   360,   140,     5,    0;    720,  3600,  3000,   450,    6, 0;   5040, 37800, 54600, 18900, 1302, 7, 0; MAPLE T:= (n, k)-> (n!/k!)*Stirling2(n-1, n-k): seq(seq(T(n, k), k=1..n), n=1..10);  # Alois P. Heinz, Nov 13 2013 MATHEMATICA Table[Table[n!/k! StirlingS2[n-1, n-k], {k, 1, n}], {n, 0, 10}]//Grid  (* Geoffrey Critzer, Dec 01 2012 *) PROG (PARI) A055302(n, k)=n!/k!*stirling(n-1, n-k, 2); for(n=1, 10, for(k=1, n, print1(A055302(n, k), ", ")); print()); \\ Joerg Arndt, Oct 27 2014 CROSSREFS Row sums give A000169. Columns 1 through 12: A000142, A055303-A055313. Cf. A055314. Cf. A248120 for a natural refinement. Sequence in context: A269795 A095834 A106828 * A055349 A161174 A291240 Adjacent sequences:  A055299 A055300 A055301 * A055303 A055304 A055305 KEYWORD nonn,tabl,eigen AUTHOR Christian G. Bower, May 11 2000 STATUS approved

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Last modified July 21 06:55 EDT 2019. Contains 325192 sequences. (Running on oeis4.)