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A236396
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Triangle read by rows: T(n,k) = number of rooted labeled trees with n nodes and height <= k, for n >= 1, 0 <= k <= n-1.
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5
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1, 0, 2, 0, 3, 9, 0, 4, 40, 64, 0, 5, 205, 505, 625, 0, 6, 1176, 4536, 7056, 7776, 0, 7, 7399, 46249, 89929, 112609, 117649, 0, 8, 50576, 526352, 1284032, 1835072, 2056832, 2097152, 0, 9, 372537, 6604497, 20351601, 33188481, 40325121, 42683841, 43046721
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history;
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OFFSET
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1,3
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COMMENTS
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If we replace each row by its differences we get A034855.
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LINKS
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Alois P. Heinz, Rows n = 1..100, flattened
J. Riordan, Enumeration of trees by height and diameter, IBM J. Res. Dev. 4 (1960), 473-478.
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EXAMPLE
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Triangle begins:
[1],
[0, 2],
[0, 3, 9],
[0, 4, 40, 64],
[0, 5, 205, 505, 625],
[0, 6, 1176, 4536, 7056, 7776],
[0, 7, 7399, 46249, 89929, 112609, 117649],
[0, 8, 50576, 526352, 1284032, 1835072, 2056832, 2097152],
...
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MAPLE
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gf:= proc(k) gf(k):= `if`(k=0, x, x*exp(gf(k-1))) end:
A:= proc(n, k) A(n, k):= n!*coeff(series(gf(k), x, n+1), x, n) end:
[seq ([seq (A(n, d), d=0..n-1)], n=1..12)];
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MATHEMATICA
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gf[k_] := gf[k] = If[k == 0, x, x*E^gf[k-1]]; a[n_, k_] := n!*Coefficient[Series[gf[k], {x, 0, n+1}], x, n]; Table[Table[a[n, d], {d, 0, n-1}], {n, 1, 12}] // Flatten (* Jean-François Alcover, Mar 07 2014, after Maple *)
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CROSSREFS
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Cf. A034855, A001854, A235595, A234953.
Sequence in context: A074104 A071411 A255384 * A251592 A121065 A077928
Adjacent sequences: A236393 A236394 A236395 * A236397 A236398 A236399
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KEYWORD
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nonn,tabl
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AUTHOR
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N. J. A. Sloane, Jan 28 2014
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STATUS
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approved
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