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 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, 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,9 LINKS M. Bousquet-Mélou, Limit laws for embedded trees, arXiv:math/0501266 [math.CO], 2005. FORMULA G.f. of k-th row: A(t) = B(t)*(1-C(t)^(k+2))*(1-C(t)^(k+7))/((1-C(t)^(k+4))*(1-C(t)^(k+5))), with B(t) the g.f. of A000108 and C(t) the g.f. of A101490. EXAMPLE 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, ... 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 *) CROSSREFS Rows converge to A000108. First row is A101488. Cf. A101490. Sequence in context: A135229 A257543 A081372 * A104156 A070166 A131373 Adjacent sequences:  A101486 A101487 A101488 * A101490 A101491 A101492 KEYWORD nonn,tabl AUTHOR Ralf Stephan, Jan 21 2005 STATUS approved

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Last modified September 24 01:21 EDT 2020. Contains 337315 sequences. (Running on oeis4.)