A143911 - OEIS

A143911

Triangle read by rows: T(n,k) = number of forests on n labeled nodes, where k is the maximum of the number of edges per tree (n>=1, 0<=k<=n-1).

1, 1, 1, 1, 3, 3, 1, 9, 12, 16, 1, 25, 60, 80, 125, 1, 75, 330, 480, 750, 1296, 1, 231, 1680, 3920, 5250, 9072, 16807, 1, 763, 9408, 33600, 49000, 72576, 134456, 262144, 1, 2619, 56952, 254016, 598500, 762048, 1210104, 2359296, 4782969, 1, 9495, 348120

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,5

LINKS

Alois P. Heinz, Rows n = 1..141, flattened

FORMULA

See program.

EXAMPLE

T(4,1) = 9, because 9 forests on 4 labeled nodes have 1 as the maximum of the number of edges per tree:

.1-2. .1.2. .1.2. .1.2. .1.2. .1.2. .1-2. .1.2. .1.2.

..... ...|. ..... .|... ..\.. ../.. ..... .|.|. ..X..

.4.3. .4.3. .4-3. .4.3. .4.3. .4.3. .4-3. .4.3. .4.3.

Triangle begins:

1, 1;

1, 3, 3;

1, 9, 12, 16;

1, 25, 60, 80, 125;

1, 75, 330, 480, 750, 1296;

MAPLE

A:= (n, k)-> coeff(series(exp(add(j^(j-2) *x^j/j!, j=1..k)), x, n+1), x, n)*n!: T:= (n, k)-> A(n, k+1)-A(n, k): seq(seq(T(n, k), k=0..n-1), n=1..11);

MATHEMATICA

A[n_, k_] := SeriesCoefficient[Exp[Sum[j^(j-2)*x^j/j!, {j, 1, k}]], {x, 0, n}]*n!; T[n_, k_] := A[n, k+1] - A[n, k];

Table[T[n, k], {n, 1, 11}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, May 31 2016, translated from Maple *)

CROSSREFS

Columns k=0-1 give: A000012, A001189.

Row sums give A001858.

Rightmost diagonal gives A000272.

Cf. A138464.

Sequence in context: A260301 A216916 A157401 * A185422 A131889 A292386

Adjacent sequences: A143908 A143909 A143910 * A143912 A143913 A143914

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Sep 04 2008

STATUS

approved