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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
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
KEYWORD
nonn,tabl
AUTHOR
Ralf Stephan, Jan 21 2005
STATUS
approved