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A292085 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.

%I A292085 #18 Sep 07 2018 17:01:46
%S A292085 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 A292085 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 A292085 33,87,228,426,98,0,1,1,2,5,12,33,89,251,656,1194,207,0
%N A292085 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 A292085 Alois P. Heinz, <a href="/A292085/b292085.txt">Antidiagonals n = 1..141, flattened</a>
%H A292085 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F A292085 A(n,k) = Sum_{j=1..k} A292086(n,j).
%e A292085 :               T(4,3) = 4             :
%e A292085 :                                      :
%e A292085 :       o       o         o       o    :
%e A292085 :      / \     / \       / \     /|\   :
%e A292085 :     o   N   o   o     o   N   o N N  :
%e A292085 :    / \     ( ) ( )   /|\     ( )     :
%e A292085 :   o   N    N N N N  N N N    N N     :
%e A292085 :  ( )                                 :
%e A292085 :  N N                                 :
%e A292085 :                                      :
%e A292085 Square array A(n,k) begins:
%e A292085   1,  1,   1,   1,   1,   1,   1,   1, ...
%e A292085   0,  1,   1,   1,   1,   1,   1,   1, ...
%e A292085   0,  1,   2,   2,   2,   2,   2,   2, ...
%e A292085   0,  2,   4,   5,   5,   5,   5,   5, ...
%e A292085   0,  3,   9,  11,  12,  12,  12,  12, ...
%e A292085   0,  6,  23,  30,  32,  33,  33,  33, ...
%e A292085   0, 11,  58,  80,  87,  89,  90,  90, ...
%e A292085   0, 23, 156, 228, 251, 258, 260, 261, ...
%p A292085 b:= proc(n, i, v, k) option remember; `if`(n=0,
%p A292085       `if`(v=0, 1, 0), `if`(i<1 or v<1 or n<v, 0,
%p A292085       `if`(v=n, 1, add(binomial(A(i,k)+j-1, j)*
%p A292085        b(n-i*j, i-1, v-j, k), j=0..min(n/i, v)))))
%p A292085     end:
%p A292085 A:= proc(n, k) option remember; `if`(n<2, n,
%p A292085       add(b(n, n+1-j, j, k), j=2..min(n, k)))
%p A292085     end:
%p A292085 seq(seq(A(n, 1+d-n), n=1..d), d=1..14);
%t A292085 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 A292085 A[n_, k_] := A[n, k] = If[n < 2, n, Sum[b[n, n + 1 - j, j, k], {j, 2, Min[n, k]}]];
%t A292085 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 A292085 Columns k=1-10 give: A063524, A001190, A268172, A292210, A292211, A292212, A292213, A292214, A292215, A292216.
%Y A292085 Main diagonal gives A000669.
%Y A292085 Cf. A244372, A288942, A292086.
%K A292085 nonn,tabl
%O A292085 1,13
%A A292085 _Alois P. Heinz_, Sep 08 2017