A307214 Triangle read by rows: number of forests of (k+1) ordered trees with 2(n-k) edges having root of even degree and nonroot nodes of outdegree 0 or 2.

1, 1, 1, 3, 2, 1, 10, 7, 3, 1, 35, 26, 12, 4, 1, 126, 99, 49, 18, 5, 1, 462, 382, 201, 80, 25, 6, 1, 1716, 1486, 825, 351, 120, 33, 7, 1, 6435, 5812, 3384, 1524, 560, 170, 42, 8, 1, 24310, 22819, 13866, 6562, 2561, 840, 231, 52, 9, 1

Offset

0,4

Links

Table of n, a(n) for n=0..54.

Formula

G.f.: (2*x)/(-2*x^2*y+4*x+sqrt(1-4*x)-1).

T(n,m) = Sum_{k=1..n-m} k/(n-m) * C(2*n-2*m,n-m-k) * C(k+m,k).

Example

1;
1, 1;
3, 2, 1;
10, 7, 3, 1;
126, 99, 49, 18, 5, 1;
Forests have 4 edges and 2 trees. T(3,1)=7.
|...........|.....x.x.|....x.x...|
|....x.x.x.x|....x.o..|.....o.x..|
|.r.....r...|.r...r...|.r....r...|
.................................
        |.x.x..x.x.|
        | .r....r..|
''''''''''''''''''''''''''''''''''
|...........|..x.x....|.x.x......|
|.x.x.x.x...|.x.o.....|..o.x.....|
|....r....r.|..r....r.|...r....r.|

Prog

(Maxima) T(n, m):=if n=m then 1 else sum(k/(n-m)*binomial(2*n-2*m, n-m-k)*binomial(k+m, k), k, 1, n-m);

Crossrefs

Cf. A088218.

Sequence in context: A267019 A100100 A248036 * A185967 A188111 A102472

Adjacent sequences: A307211 A307212 A307213 * A307215 A307216 A307217

Keyword

nonn,tabl

Author

Vladimir Kruchinin, Mar 28 2019

Status

approved