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Number A(n,k) of (unlabeled) rooted trees with n leaf nodes and without unary nodes or outdegrees larger than k; square array A(n,k), n>=0, k>=0, read by antidiagonals.
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%I #18 Sep 07 2018 17:01:46

%S 1,1,0,1,1,0,1,1,1,0,1,1,2,2,0,1,1,2,4,3,0,1,1,2,5,9,6,0,1,1,2,5,11,

%T 23,11,0,1,1,2,5,12,30,58,23,0,1,1,2,5,12,32,80,156,46,0,1,1,2,5,12,

%U 33,87,228,426,98,0,1,1,2,5,12,33,89,251,656,1194,207,0

%N Number A(n,k) of (unlabeled) rooted trees with n leaf nodes and without unary nodes or outdegrees larger than k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%H Alois P. Heinz, <a href="/A292085/b292085.txt">Antidiagonals n = 1..141, flattened</a>

%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>

%F A(n,k) = Sum_{j=1..k} A292086(n,j).

%e : T(4,3) = 4 :

%e : :

%e : o o o o :

%e : / \ / \ / \ /|\ :

%e : o N o o o N o N N :

%e : / \ ( ) ( ) /|\ ( ) :

%e : o N N N N N N N N N N :

%e : ( ) :

%e : N N :

%e : :

%e Square array A(n,k) begins:

%e 1, 1, 1, 1, 1, 1, 1, 1, ...

%e 0, 1, 1, 1, 1, 1, 1, 1, ...

%e 0, 1, 2, 2, 2, 2, 2, 2, ...

%e 0, 2, 4, 5, 5, 5, 5, 5, ...

%e 0, 3, 9, 11, 12, 12, 12, 12, ...

%e 0, 6, 23, 30, 32, 33, 33, 33, ...

%e 0, 11, 58, 80, 87, 89, 90, 90, ...

%e 0, 23, 156, 228, 251, 258, 260, 261, ...

%p b:= proc(n, i, v, k) option remember; `if`(n=0,

%p `if`(v=0, 1, 0), `if`(i<1 or v<1 or n<v, 0,

%p `if`(v=n, 1, add(binomial(A(i,k)+j-1, j)*

%p b(n-i*j, i-1, v-j, k), j=0..min(n/i, v)))))

%p end:

%p A:= proc(n, k) option remember; `if`(n<2, n,

%p add(b(n, n+1-j, j, k), j=2..min(n, k)))

%p end:

%p seq(seq(A(n, 1+d-n), n=1..d), d=1..14);

%t b[n_, i_, v_, k_] := b[n, i, v, k] = If[n == 0, If[v == 0, 1, 0], If[i < 1 || v < 1 || n < v, 0, If[v == n, 1, Sum[Binomial[A[i, k] + j - 1, j]*b[n - i*j, i - 1, v - j, k], {j, 0, Min[n/i, v]}]]]];

%t A[n_, k_] := A[n, k] = If[n < 2, n, Sum[b[n, n + 1 - j, j, k], {j, 2, Min[n, k]}]];

%t Table[Table[A[n, 1 + d - n], {n, 1, d}], {d, 1, 14}] // Flatten (* _Jean-François Alcover_, Nov 07 2017, after _Alois P. Heinz_ *)

%Y Columns k=1-10 give: A063524, A001190, A268172, A292210, A292211, A292212, A292213, A292214, A292215, A292216.

%Y Main diagonal gives A000669.

%Y Cf. A244372, A288942, A292086.

%K nonn,tabl

%O 1,13

%A _Alois P. Heinz_, Sep 08 2017