cp's OEIS Frontend

A255636 Number A(n,k) of n-node rooted trees with a forbidden limb of length k; square array A(n,k), n>=1, k>=1, read by antidiagonals.

%I A255636 #20 Sep 05 2018 12:27:59
%S A255636 1,1,0,1,1,0,1,1,1,0,1,1,2,2,0,1,1,2,3,4,0,1,1,2,4,7,8,0,1,1,2,4,8,15,
%T A255636 17,0,1,1,2,4,9,18,35,36,0,1,1,2,4,9,19,43,81,79,0,1,1,2,4,9,20,46,
%U A255636 102,195,175,0,1,1,2,4,9,20,47,110,251,473,395,0
%N A255636 Number A(n,k) of n-node rooted trees with a forbidden limb of length k; square array A(n,k), n>=1, k>=1, read by antidiagonals.
%C A255636 Any rootward k-node path starting at a leaf contains the root or a branching node.
%H A255636 Alois P. Heinz, <a href="/A255636/b255636.txt">Antidiagonals n = 1..141, flattened</a>
%e A255636 :    o      o        o      o    o       o      o    o
%e A255636 :  /(|)\    |       / \    /|\   |       |     / \   |
%e A255636 : o ooo o   o      o   o  o o o  o       o    o   o  o
%e A255636 :         /( )\   /|\    / \     |      / \   |      |
%e A255636 :        o o o o o o o  o   o    o     o   o  o      o
%e A255636 :                               /|\   / \    / \     |
%e A255636 :                              o o o o   o  o   o    o
%e A255636 : A(6,2) = 8                                        / \
%e A255636 :                                                  o   o
%e A255636 Square array A(n,k) begins:
%e A255636   1,   1,   1,   1,   1,   1,   1,   1,   1,   1, ...
%e A255636   0,   1,   1,   1,   1,   1,   1,   1,   1,   1, ...
%e A255636   0,   1,   2,   2,   2,   2,   2,   2,   2,   2, ...
%e A255636   0,   2,   3,   4,   4,   4,   4,   4,   4,   4, ...
%e A255636   0,   4,   7,   8,   9,   9,   9,   9,   9,   9, ...
%e A255636   0,   8,  15,  18,  19,  20,  20,  20,  20,  20, ...
%e A255636   0,  17,  35,  43,  46,  47,  48,  48,  48,  48, ...
%e A255636   0,  36,  81, 102, 110, 113, 114, 115, 115, 115, ...
%e A255636   0,  79, 195, 251, 273, 281, 284, 285, 286, 286, ...
%e A255636   0, 175, 473, 625, 684, 706, 714, 717, 718, 719, ...
%p A255636 with(numtheory):
%p A255636 g:= proc(n, k) option remember; `if`(n=0, 1, add(add(d*(g(d-1, k)-
%p A255636       `if`(d=k, 1, 0)), d=divisors(j))*g(n-j, k), j=1..n)/n)
%p A255636     end:
%p A255636 A:= (n, k)-> g(n-1, k):
%p A255636 seq(seq(A(n, 1+d-n), n=1..d), d=1..14);
%t A255636 g[n_, k_] := g[n, k] = If[n == 0, 1, Sum[Sum[d*(g[d - 1, k] - If[d == k, 1, 0]), {d, Divisors[j]}]*g[n - j, k], {j, 1, n}]/n]; A[n_, k_] := g[n - 1, k]; Table[Table[A[n, 1 + d - n], {n, 1, d}], {d, 1, 14}] // Flatten (* _Jean-François Alcover_, Feb 22 2016, after _Alois P. Heinz_ *)
%Y A255636 Columns k=1-10 give: A063524, A002955, A052321, A052327, A052328, A052329, A255637, A255638, A255639, A255640.
%Y A255636 Main diagonal gives A000081.
%Y A255636 Cf. A255704.
%K A255636 nonn,tabl
%O A255636 1,13
%A A255636 _Alois P. Heinz_, Feb 28 2015