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A210725
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Triangle read by rows: T(n,k) = number of forests of labeled rooted trees with n nodes and height at most k (n>=1, 0<=k<=n-1).
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7
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1, 1, 3, 1, 10, 16, 1, 41, 101, 125, 1, 196, 756, 1176, 1296, 1, 1057, 6607, 12847, 16087, 16807, 1, 6322, 65794, 160504, 229384, 257104, 262144, 1, 41393, 733833, 2261289, 3687609, 4480569, 4742649, 4782969, 1, 293608, 9046648, 35464816, 66025360, 87238720, 96915520, 99637120, 100000000
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graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,3
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LINKS
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EXAMPLE
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Triangle begins:
1;
1, 3;
1, 10, 16;
1, 41, 101, 125;
1, 196, 756, 1176, 1296;
1, 1057, 6607, 12847, 16087, 16807;
...
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MAPLE
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f:= proc(k) f(k):= `if`(k<0, 1, exp(x*f(k-1))) end:
T:= (n, k)-> coeff(series(f(k), x, n+1), x, n) *n!:
# second Maple program:
T:= proc(n, h) option remember; `if`(min(n, h)=0, 1, add(
binomial(n-1, j-1)*j*T(j-1, h-1)*T(n-j, h), j=1..n))
end:
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MATHEMATICA
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f[_?Negative] = 1; f[k_] := Exp[x*f[k-1]]; t[n_, k_] := Coefficient[Series[f[k], {x, 0, n+1}], x, n]*n!; Table[Table[t[n, k], {k, 0, n-1}], {n, 1, 9}] // Flatten (* Jean-François Alcover, Oct 30 2013, after Maple *)
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PROG
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(Python)
from sympy.core.cache import cacheit
from sympy import binomial
@cacheit
def T(n, h): return 1 if min(n, h)==0 else sum([binomial(n - 1, j - 1)*j*T(j - 1, h - 1)*T(n - j, h) for j in range(1, n + 1)])
for n in range(1, 11): print([T(n, k) for k in range(n)]) # Indranil Ghosh, Aug 21 2017, after second Maple code
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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