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A255517 Number A(n,k) of rooted identity trees with n nodes and k-colored non-root nodes; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A255517 #40 Sep 07 2018 15:07:56
%S A255517 0,0,1,0,1,0,0,1,1,0,0,1,2,1,0,0,1,3,5,2,0,0,1,4,12,18,3,0,0,1,5,22,
%T A255517 64,66,6,0,0,1,6,35,156,363,266,12,0,0,1,7,51,310,1193,2214,1111,25,0,
%U A255517 0,1,8,70,542,2980,9748,14043,4792,52,0,0,1,9,92,868,6273,30526,82916,91857,21124,113,0
%N A255517 Number A(n,k) of rooted identity trees with n nodes and k-colored non-root nodes; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%C A255517 From _Vaclav Kotesovec_, Feb 24 2015: (Start)
%C A255517 k    Limit n->infinity A(n,k)^(1/n)
%C A255517 1    2.517540352632003890795354598463447277335981266803... = A246169
%C A255517 2    5.249032491228170579164952216184309265343086337648... = A246312
%C A255517 3    7.969494030514425004826375511986491746399264355846...
%C A255517 4   10.688492754969652458452048798468242930479212456958...
%C A255517 5   13.407087472537747579787047072702638639945914705837...
%C A255517 6   16.125529360448558670505097146631763969697822205298...
%C A255517 7   18.843901825822305757579605844910623225182677164912...
%C A255517 8   21.562238702430237066018783115405680041128676137631...
%C A255517 9   24.280555694806692616578932533497629224907619468796...
%C A255517 10  26.998860838916733933849490675388336975888308433826...
%C A255517 100 271.64425688361559470587959030374804709717287744789...
%C A255517 Conjecture: For big k the limit asymptotically approaches k*exp(1).
%C A255517 (End)
%H A255517 Alois P. Heinz, <a href="/A255517/b255517.txt">Antidiagonals n = 0..140, flattened</a>
%e A255517 A(3,2) = 5:
%e A255517   o    o    o    o      o
%e A255517   |    |    |    |     / \
%e A255517   1    1    2    2    1   2
%e A255517   |    |    |    |
%e A255517   1    2    1    2
%e A255517 Square array A(n,k) begins:
%e A255517   0,  0,   0,    0,    0,     0,     0, ...
%e A255517   1,  1,   1,    1,    1,     1,     1, ...
%e A255517   0,  1,   2,    3,    4,     5,     6, ...
%e A255517   0,  1,   5,   12,   22,    35,    51, ...
%e A255517   0,  2,  18,   64,  156,   310,   542, ...
%e A255517   0,  3,  66,  363, 1193,  2980,  6273, ...
%e A255517   0,  6, 266, 2214, 9748, 30526, 77262, ...
%p A255517 with(numtheory):
%p A255517 A:= proc(n, k) option remember; `if`(n<2, n, add(A(n-j, k)*add(
%p A255517       k*A(d, k)*d*(-1)^(j/d+1), d=divisors(j)), j=1..n-1)/(n-1))
%p A255517     end:
%p A255517 seq(seq(A(n, d-n), n=0..d), d=0..14);
%t A255517 A[n_, k_] := A[n, k] = If[n<2, n, Sum[A[n-j, k]*Sum[k*A[d, k]*d*(-1)^(j/d + 1), {d, Divisors[j]}], {j, 1, n-1}]/(n-1)]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* _Jean-François Alcover_, Feb 22 2016, after _Alois P. Heinz_ *)
%Y A255517 Columns k=1-10 give: A004111, A005753, A052757, A052772, A052797, A255518, A255519, A255520, A255521, A255522.
%Y A255517 Rows n=0-4 give: A000004, A000012, A001477, A000326, 2*A051662(k-1) for k>0.
%Y A255517 Lower diagonal gives A255523.
%Y A255517 Cf. A242249, A256068.
%K A255517 nonn,tabl
%O A255517 0,13
%A A255517 _Alois P. Heinz_, Feb 24 2015