A101489 Square array T(n,k), read by antidiagonals: number of binary trees, with n nodes that have no label greater than k.
1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 4, 4, 1, 1, 2, 5, 10, 10, 1, 1, 2, 5, 13, 26, 26, 1, 1, 2, 5, 14, 37, 73, 73, 1, 1, 2, 5, 14, 41, 109, 213, 213, 1, 1, 2, 5, 14, 42, 126, 334, 645, 645, 1, 1, 2, 5, 14, 42, 131, 398, 1050, 2007, 2007, 1, 1, 2, 5, 14, 42, 132, 422, 1289, 3377, 6391, 6391
Offset: 0
Examples
1, 1, 1, 2, 4, 10, 26, 73, 213, 645, ... 1, 1, 2, 4, 10, 26, 73, 213, 645, 2007, ... 1, 1, 2, 5, 13, 37, 109, 334, 1050, 3377, ... 1, 1, 2, 5, 14, 41, 126, 398, 1289, 4253, ... 1, 1, 2, 5, 14, 42, 131, 422, 1390, 4664, ... 1, 1, 2, 5, 14, 42, 132, 428, 1422, 4812, ... 1, 1, 2, 5, 14, 42, 132, 429, 1429, 4853, ... 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4861, ...
Links
- M. Bousquet-Mélou, Limit laws for embedded trees, arXiv:math/0501266 [math.CO], 2005.
Programs
-
Mathematica
nmax = 11; b[t_] = Sum[Binomial[2n, n]/(n + 1) t^n, {n, 0, nmax}] ; c[t_] = 1; Do[c[t_] = t (1 + c[t]^2)^2/(1 - c[t] + c[t]^2) + O[t]^(nmax + 1), {nmax + 1}]; a[n_, t_] := a[n, t] = b[t] (1 - c[t]^(n + 2)) ((1 - c[t]^(n + 7))/((1 - c[t]^(n + 4)) (1 - c[t]^(n + 5)))) + O[t]^(nmax + 1); T[n_, k_] := SeriesCoefficient[a[n, t], {t, 0, k}]; Table[T[n - k, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 25 2018 *)