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%I #12 Jul 25 2018 03:47:03
%S 1,1,1,1,2,3,1,2,7,12,1,2,8,31,56,1,2,8,39,156,288,1,2,8,40,211,851,
%T 1584,1,2,8,40,223,1219,4909,9152,1,2,8,40,224,1327,7371,29506,54912,
%U 1,2,8,40,224,1343,8250,46099,183043,339456,1,2,8,40,224,1344,8427,52938,295915,1164387,2149888
%N Square array T(n,k), read by antidiagonals: number of labeled trees, with increments of labels along edges constrained to +-1, with n nodes that have no label greater than k.
%H M. Bousquet-Mélou, <a href="https://arxiv.org/abs/math/0501266">Limit laws for embedded trees</a>, arXiv:math/0501266 [math.CO], 2005.
%F G.f. of k-th row: A(t)=B(t)*(1-C(t)^(k+1))*(1-C(t)^(k+5))/[(1-C(t)^(k+2))*(1-C(t)^(k+4))], with tB(t) the g.f. of A052701 and C(t) the g.f. of A101478.
%e 1, 1, 3, 12, 56, 288, 1584, 9152, 54912, 339456, ...
%e 1, 2, 7, 31, 156, 851, 4909, 29506, 183043, 1164387, ...
%e 1, 2, 8, 39, 211, 1219, 7371, 46099, 295915, 1939395, ...
%e 1, 2, 8, 40, 223, 1327, 8250, 52938, 347941, 2330532, ...
%e 1, 2, 8, 40, 224, 1343, 8427, 54625, 362833, 2456261, ...
%e 1, 2, 8, 40, 224, 1344, 8447, 54887, 365688, 2484384, ...
%e 1, 2, 8, 40, 224, 1344, 8448, 54911, 366051, 2488831, ...
%e 1, 2, 8, 40, 224, 1344, 8448, 54912, 366079, 2489311, ...
%e 1, 2, 8, 40, 224, 1344, 8448, 54912, 366080, 2489343, ...
%e 1, 2, 8, 40, 224, 1344, 8448, 54912, 366080, 2489344, ...
%t nmax = 11;
%t b[x_] = Sum[2^(n - 1)*(2*n - 2)!/(n - 1)!/n! x^n, {n, 1, nmax}];
%t c[x_] = 0; Do[c[x_] = x*(1 + c[x])^4/(1 + c[x]^2) + O[x]^nmax, {nmax}];
%t a[n_, t_] := a[n, t] = b[t]*(1 - c[t]^(n + 1))*(1 - c[t]^(n + 5))/((1 - c[t]^(n + 2))*(1 - c[t]^(n + 4)));
%t T[n_, k_] := SeriesCoefficient[a[n, t], {t, 0, k}];
%t Table[T[n - k, k], {n, 1, nmax}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jul 25 2018 *)
%Y Rows converge to A052701. First row is A000257.
%Y Cf. A052701, A101478.
%K nonn,tabl
%O 0,5
%A _Ralf Stephan_, Jan 21 2005
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