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