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A290353
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Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the k-th Euler transform of the sequence with g.f. 1+x.
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22
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1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 3, 3, 1, 0, 1, 1, 4, 6, 5, 1, 0, 1, 1, 5, 10, 14, 7, 1, 0, 1, 1, 6, 15, 30, 27, 11, 1, 0, 1, 1, 7, 21, 55, 75, 58, 15, 1, 0, 1, 1, 8, 28, 91, 170, 206, 111, 22, 1, 0, 1, 1, 9, 36, 140, 336, 571, 518, 223, 30, 1, 0
(list;
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refs;
listen;
history;
text;
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OFFSET
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0,13
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COMMENTS
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A(n,k) is the number of unlabeled rooted trees with exactly n leaves, all in level k. A(3,3) = 6:
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LINKS
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B. A. Huberman and T. Hogg, Complexity and adaptation, Evolution, games and learning (Los Alamos, N.M., 1985). Phys. D 22 (1986), no. 1-3, 376-384.
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FORMULA
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G.f. of column k=0: 1+x, of column k>0: Product_{j>0} 1/(1-x^j)^A(j,k-1).
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EXAMPLE
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Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, 6, 7, 8, ...
0, 1, 3, 6, 10, 15, 21, 28, 36, ...
0, 1, 5, 14, 30, 55, 91, 140, 204, ...
0, 1, 7, 27, 75, 170, 336, 602, 1002, ...
0, 1, 11, 58, 206, 571, 1337, 2772, 5244, ...
0, 1, 15, 111, 518, 1789, 5026, 12166, 26328, ...
0, 1, 22, 223, 1344, 5727, 19193, 54046, 133476, ...
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MAPLE
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with(numtheory):
A:= proc(n, k) option remember; `if`(n<2, 1, `if`(k=0, 0, add(
add(A(d, k-1)*d, d=divisors(j))*A(n-j, k), j=1..n)/n))
end:
seq(seq(A(n, d-n), n=0..d), d=0..14);
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MATHEMATICA
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A[n_, k_]:=b[n, k]=If[n<2, 1, If[k==0, 0, Sum[Sum[A[d, k - 1]*d, {d, Divisors[j]}] A[n - j, k], {j, n}]/n]]; Table[A[n, d - n], {d, 0, 14}, {n, 0, d}]//Flatten (* Indranil Ghosh, Jul 30 2017, after Maple code *)
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CROSSREFS
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Rows 0+1,2-10 give: A000012, A001477, A000217, A000330, A007715, A290360, A290361, A290362, A290363, A290364.
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KEYWORD
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AUTHOR
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STATUS
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approved
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