A307214 - OEIS

%I #16 Apr 01 2019 08:30:07

%S 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,

%T 6,1,1716,1486,825,351,120,33,7,1,6435,5812,3384,1524,560,170,42,8,1,

%U 24310,22819,13866,6562,2561,840,231,52,9,1

%N 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.

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

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

%e 1;

%e 1, 1;

%e 3, 2, 1;

%e 10, 7, 3, 1;

%e 126, 99, 49, 18, 5, 1;

%e Forests have 4 edges and 2 trees. T(3,1)=7.

%e |...........|.....x.x.|....x.x...|

%e |....x.x.x.x|....x.o..|.....o.x..|

%e |.r.....r...|.r...r...|.r....r...|

%e .................................

%e         |.x.x..x.x.|

%e         | .r....r..|

%e ''''''''''''''''''''''''''''''''''

%e |...........|..x.x....|.x.x......|

%e |.x.x.x.x...|.x.o.....|..o.x.....|

%e |....r....r.|..r....r.|...r....r.|

%o (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);

%Y Cf. A088218.

%K nonn,tabl

%O 0,4

%A _Vladimir Kruchinin_, Mar 28 2019