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 A210725 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). 7
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 LINKS Alois P. Heinz, Rows n = 1..141, flattened J. Riordan, Forests of labeled trees, J. Combin. Theory, 5 (1968), 90-103. EXAMPLE Triangle begins:   1;   1,    3;   1,   10,   16;   1,   41,  101,   125;   1,  196,  756,  1176,  1296;   1, 1057, 6607, 12847, 16087, 16807;   ... MAPLE 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!: seq(seq(T(n, k), k=0..n-1), n=1..9); # Alois P. Heinz, May 30 2012 # 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: seq(seq(T(n, k), k=0..n-1), n=1..10);  # Alois P. Heinz, Aug 21 2017 MATHEMATICA 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 *) PROG (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 CROSSREFS Diagonals include A000248, A000949, A000950, A000951, A000272. Sequence in context: A225725 A095327 A225753 * A048953 A200652 A276391 Adjacent sequences:  A210722 A210723 A210724 * A210726 A210727 A210728 KEYWORD nonn,tabl AUTHOR N. J. A. Sloane, May 09 2012 STATUS approved

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Last modified April 18 03:59 EDT 2021. Contains 343072 sequences. (Running on oeis4.)