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A292086 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.

%I A292086 #24 Mar 13 2024 10:30:07
%S A292086 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 A292086 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 A292086 207,3186,2442,861,261,78,23,7,2,1,0,451,9351,8006,2897,887,262,78,23,7,2,1
%N A292086 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 A292086 Alois P. Heinz, <a href="/A292086/b292086.txt">Rows n = 1..141, flattened</a>
%H A292086 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F A292086 T(n,k) = A292085(n,k) - A292085(n,k-1) for k>2, T(n,1) = A292085(n,1).
%e A292086 :   T(4,2) = 2        :   T(4,3) = 2      : T(4,4) = 1 :
%e A292086 :                     :                   :            :
%e A292086 :       o       o     :      o       o    :     o      :
%e A292086 :      / \     / \    :     / \     /|\   :   /( )\    :
%e A292086 :     o   N   o   o   :    o   N   o N N  :  N N N N   :
%e A292086 :    / \     ( ) ( )  :   /|\     ( )     :            :
%e A292086 :   o   N    N N N N  :  N N N    N N     :            :
%e A292086 :  ( )                :                   :            :
%e A292086 :  N N                :                   :            :
%e A292086 :                     :                   :            :
%e A292086 Triangle T(n,k) begins:
%e A292086   1;
%e A292086   0,  1;
%e A292086   0,  1,   1;
%e A292086   0,  2,   2,   1;
%e A292086   0,  3,   6,   2,  1;
%e A292086   0,  6,  17,   7,  2,  1;
%e A292086   0, 11,  47,  22,  7,  2, 1;
%e A292086   0, 23, 133,  72, 23,  7, 2, 1;
%e A292086   0, 46, 380, 230, 77, 23, 7, 2, 1;
%e A292086   ...
%p A292086 b:= proc(n, i, v, k) option remember; `if`(n=0,
%p A292086       `if`(v=0, 1, 0), `if`(i<1 or v<1 or n<v, 0,
%p A292086       `if`(v=n, 1, add(binomial(A(i,k)+j-1, j)*
%p A292086        b(n-i*j, i-1, v-j, k), j=0..min(n/i, v)))))
%p A292086     end:
%p A292086 A:= proc(n, k) option remember; `if`(n<2, n,
%p A292086       add(b(n, n+1-j, j, k), j=2..min(n, k)))
%p A292086     end:
%p A292086 T:= (n, k)-> A(n, k)-`if`(k=1, 0, A(n, k-1)):
%p A292086 seq(seq(T(n, k), k=1..n), n=1..15);
%t A292086 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 A292086 A[n_, k_] := A[n, k] = If[n < 2, n, Sum[b[n, n + 1 - j, j, k], {j, 2, Min[n, k]}]];
%t A292086 T[n_, k_] := A[n, k] - If[k == 1, 0, A[n, k - 1]];
%t A292086 Table[Table[T[n, k], {k, 1, n}], {n, 1, 15}] // Flatten (* _Jean-François Alcover_, Nov 07 2017, after _Alois P. Heinz_ *)
%Y A292086 Columns k=1-10 give: A063524, A001190 (for n>1), A292229, A292230, A292231, A292232, A292233, A292234, A292235, A292236.
%Y A292086 Row sums give A000669.
%Y A292086 Limit of reversed rows gives A292087.
%Y A292086 Cf. A244372, A288942, A292085.
%K A292086 nonn,tabl
%O A292086 1,8
%A A292086 _Alois P. Heinz_, Sep 08 2017