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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.

%I A101489 #10 Jul 25 2018 03:47:34
%S A101489 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,
%T A101489 14,37,73,73,1,1,2,5,14,41,109,213,213,1,1,2,5,14,42,126,334,645,645,
%U A101489 1,1,2,5,14,42,131,398,1050,2007,2007,1,1,2,5,14,42,132,422,1289,3377,6391,6391
%N A101489 Square array T(n,k), read by antidiagonals: number of binary trees, with n nodes that have no label greater than k.
%H A101489 M. Bousquet-Mélou, <a href="https://arxiv.org/abs/math/0501266">Limit laws for embedded trees</a>, arXiv:math/0501266 [math.CO], 2005.
%F A101489 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.
%e A101489 1, 1, 1, 2,  4, 10,  26,  73,  213,  645, ...
%e A101489 1, 1, 2, 4, 10, 26,  73, 213,  645, 2007, ...
%e A101489 1, 1, 2, 5, 13, 37, 109, 334, 1050, 3377, ...
%e A101489 1, 1, 2, 5, 14, 41, 126, 398, 1289, 4253, ...
%e A101489 1, 1, 2, 5, 14, 42, 131, 422, 1390, 4664, ...
%e A101489 1, 1, 2, 5, 14, 42, 132, 428, 1422, 4812, ...
%e A101489 1, 1, 2, 5, 14, 42, 132, 429, 1429, 4853, ...
%e A101489 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4861, ...
%t A101489 nmax = 11;
%t A101489 b[t_] = Sum[Binomial[2n, n]/(n + 1) t^n, {n, 0, nmax}] ;
%t A101489 c[t_] = 1; Do[c[t_] = t (1 + c[t]^2)^2/(1 - c[t] + c[t]^2) + O[t]^(nmax + 1), {nmax + 1}];
%t A101489 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 A101489 T[n_, k_] := SeriesCoefficient[a[n, t], {t, 0, k}];
%t A101489 Table[T[n - k, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jul 25 2018 *)
%Y A101489 Rows converge to A000108. First row is A101488.
%Y A101489 Cf. A101490.
%K A101489 nonn,tabl
%O A101489 0,9
%A A101489 _Ralf Stephan_, Jan 21 2005