A216255 Triangle read by rows: T(n,k) is the number of labeled rooted trees of height at most 2 that have exactly k nodes at a distance 2 from the root; n>=1, 0<=k<=n-1.

1, 2, 0, 3, 6, 0, 4, 24, 12, 0, 5, 60, 120, 20, 0, 6, 120, 540, 480, 30, 0, 7, 210, 1680, 3780, 1680, 42, 0, 8, 336, 4200, 17920, 22680, 5376, 56, 0, 9, 504, 9072, 63000, 161280, 122472, 16128, 72, 0, 10, 720, 17640, 181440, 787500, 1290240, 612360, 46080, 90, 0

Offset

1,2

Comments

Row sums = A052512.

Column k=1: A007531.

Links

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

Formula

E.g.f.: x*exp(x*exp(y*x)).

T(n,k) = n*C(n-1,k)*(n-k-1)^k. - Alois P. Heinz, Mar 15 2013

Example

1;
2,  0;
3,  6,   0;
4,  24,  12,    0;
5,  60,  120,   20,     0;
6,  120, 540,   480,    30,     0;
7,  210, 1680,  3780,   1680,   42,      0;
8,  336, 4200,  17920,  22680,  5376,    56,     0;
9,  504, 9072,  63000,  161280, 122472,  16128,  72,    0;
10, 720, 17640, 181440, 787500, 1290240, 612360, 46080, 90, 0;
T(4,1) = 24 because there is only one unlabeled tree on 4 nodes with exactly 1 node at distance two from the root.  It has 24 labelings.
.......o......
....../.\.....
.....o...o....
..../.........
...o..........

Maple

T:= (n, k)-> n*binomial(n-1, k)*(n-k-1)^k:
seq(seq(T(n, k), k=0..n-1), n=1..12);  # Alois P. Heinz, Mar 15 2013

Mathematica

nn=10; a=NestList[x Exp[#]&, y x, nn]; f[list_]:=Select[list, #>0&]; Map[f, Range[0, nn]!CoefficientList[Series[a[[3]], {x, 0, nn}], {x, y}]]//Grid

Crossrefs

Sequence in context: A155800 A276658 A079510 * A362788 A262256 A011120

Adjacent sequences: A216252 A216253 A216254 * A216256 A216257 A216258

Keyword

nonn,tabl

Author

Geoffrey Critzer, Mar 15 2013

Status

approved