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URL: https://oeis.org/A079268

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A079268
Triangle read by rows: d(n,k) = number of decreasing labeled trees with n nodes and largest leaf <= k, for 1 <= k <= n.
1
1, 1, 1, 1, 3, 3, 1, 7, 15, 15, 1, 15, 57, 105, 105, 1, 31, 195, 561, 945, 945, 1, 63, 633, 2685, 6555, 10395, 10395, 1, 127, 1995, 12105, 40725, 89055, 135135, 135135, 1, 255, 6177, 52605, 237555, 684495, 1381905, 2027025, 2027025, 1, 511, 18915, 223161
OFFSET
1,5
LINKS
Martin, Jeremy L. The slopes determined by n points in the plane. Duke Math. J. 131 (2006), no. 1, 119-165 (also arXiv math.AG/0302106, but beware errors).
FORMULA
Recurrence: d(n, k) = 1 for n=0 or k=1, d(n, k) = 0 for n>0 and either k<0 or k>n, d(n, k) = d(n-1, k) + d(n, k-1) + Sum_{w=0..k-2, x=0..n-k-1} binomial(k-1, w) * binomial(n-k, x) * d(n-k+w-x, w+1) * d(k-w+x, k-w-1).
EXAMPLE
Triangle begins
1,
1, 1,
1, 3, 3,
1, 7, 15, 15,
1, 15, 57, 105, 105,
1, 31, 195, 561, 945, 945,
1, 63, 633, 2685, 6555, 10395, 10395,
...
CROSSREFS
First three columns are A000012, A000225, A210448, rightmost two diagonals are both A001147, difference of second- and third-rightmost diagonals is A000165.
Sequence in context: A181304 A339435 A118408 * A102316 A261767 A355315
KEYWORD
easy,nonn,tabl
AUTHOR
Jeremy Martin (martin(AT)math.umn.edu), Feb 05 2003
STATUS
approved