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%I #24 Mar 13 2024 10:30:07
%S 1,0,1,0,1,1,0,2,2,1,0,3,6,2,1,0,6,17,7,2,1,0,11,47,22,7,2,1,0,23,133,
%T 72,23,7,2,1,0,46,380,230,77,23,7,2,1,0,98,1096,751,256,78,23,7,2,1,0,
%U 207,3186,2442,861,261,78,23,7,2,1,0,451,9351,8006,2897,887,262,78,23,7,2,1
%N Number T(n,k) of (unlabeled) rooted trees with n leaf nodes and without unary nodes such that k is the maximum of 1 and the node outdegrees; triangle T(n,k), n>=1, 1<=k<=n, read by rows.
%H Alois P. Heinz, <a href="/A292086/b292086.txt">Rows n = 1..141, flattened</a>
%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F T(n,k) = A292085(n,k) - A292085(n,k-1) for k>2, T(n,1) = A292085(n,1).
%e : T(4,2) = 2 : T(4,3) = 2 : T(4,4) = 1 :
%e : : : :
%e : o o : o o : o :
%e : / \ / \ : / \ /|\ : /( )\ :
%e : o N o o : o N o N N : N N N N :
%e : / \ ( ) ( ) : /|\ ( ) : :
%e : o N N N N N : N N N N N : :
%e : ( ) : : :
%e : N N : : :
%e : : : :
%e Triangle T(n,k) begins:
%e 1;
%e 0, 1;
%e 0, 1, 1;
%e 0, 2, 2, 1;
%e 0, 3, 6, 2, 1;
%e 0, 6, 17, 7, 2, 1;
%e 0, 11, 47, 22, 7, 2, 1;
%e 0, 23, 133, 72, 23, 7, 2, 1;
%e 0, 46, 380, 230, 77, 23, 7, 2, 1;
%e ...
%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 T:= (n, k)-> A(n, k)-`if`(k=1, 0, A(n, k-1)):
%p seq(seq(T(n, k), k=1..n), n=1..15);
%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 T[n_, k_] := A[n, k] - If[k == 1, 0, A[n, k - 1]];
%t Table[Table[T[n, k], {k, 1, n}], {n, 1, 15}] // Flatten (* _Jean-François Alcover_, Nov 07 2017, after _Alois P. Heinz_ *)
%Y Columns k=1-10 give: A063524, A001190 (for n>1), A292229, A292230, A292231, A292232, A292233, A292234, A292235, A292236.
%Y Row sums give A000669.
%Y Limit of reversed rows gives A292087.
%Y Cf. A244372, A288942, A292085.
%K nonn,tabl
%O 1,8
%A _Alois P. Heinz_, Sep 08 2017
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