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A101486 Square array T(n,k), read by antidiagonals: number of labeled trees, with increments of labels along edges constrained to -1,0,1, with n nodes that have no label greater than k.

%I A101486 #11 Jul 25 2018 03:47:11
%S A101486 1,1,2,1,3,9,1,3,17,54,1,3,18,119,378,1,3,18,134,932,2916,1,3,18,135,
%T A101486 1111,7838,24057,1,3,18,135,1133,9833,69275,208494,1,3,18,135,1134,
%U A101486 10176,90959,635279,1876446,1,3,18,135,1134,10205,95635,868827,5994584,222646205
%N A101486 Square array T(n,k), read by antidiagonals: number of labeled trees, with increments of labels along edges constrained to -1,0,1, with n nodes that have no label greater than k.
%C A101486 Rows converge to A005159.
%C A101486 First row is A000168.
%H A101486 M. Bousquet-Mélou, <a href="https://arxiv.org/abs/math/0501266">Limit laws for embedded trees</a>, arXiv:math/0501266 [math.CO], 2005.
%F A101486 G.f. of k-th row: A(t)=B(t)*(1-C(t)^(k+1))*(1-C(t)^(k+4))/[(1-C(t)^(k+2))*(1-C(t)^(k+3))], with B(t) the g.f. of A005159 and C(t) the g.f. of A101487.
%e A101486 1,2,9,54,378,2916,24057,208494,1876446,17399772,
%e A101486 1,3,17,119,932,7838,69275,635279,5994584,57872666,
%e A101486 1,3,18,134,1111,9833,90959,868827,8504314,84866778,
%e A101486 1,3,18,135,1133,10176,95635,928442,9236144,93646430,
%e A101486 1,3,18,135,1134,10205,96191,937361,9365984,95427597,
%e A101486 1,3,18,135,1134,10206,96227,938179,9381050,95673739,
%e A101486 1,3,18,135,1134,10206,96228,938222,9382179,95697199,
%e A101486 1,3,18,135,1134,10206,96228,938223,9382229,95698688,
%t A101486 nmax = 9;
%t A101486 b[t_] = 2/(1 + Sqrt[1 - 12t]) + O[t]^(nmax+1);
%t A101486 c[t_] = (1 + Sqrt[1 - 12t] - t (8 + Sqrt[2] Sqrt[(1 + Sqrt[1 - 12t] - 2 (7 + 4 Sqrt[1 - 12t]) t + 24t^2)/t^2]))/(4t) + O[t]^(nmax+1) // Simplify[#, t > 0]&;
%t A101486 a[n_, t_] := a[n, t] = b[t] (1 - c[t]^(n + 1)) (1 - c[t]^(n + 4))/((1 - c[t]^(n+2)) (1 - c[t]^(n+3))) + O[t]^(nmax+1) // Simplify[#, t > 0]&;
%t A101486 T[n_, k_] := SeriesCoefficient[a[n, t], {t, 0, k}];
%t A101486 Table[T[n - k, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jul 25 2018 *)
%Y A101486 Cf. A000168, A005159, A101487.
%K A101486 nonn,tabl
%O A101486 0,3
%A A101486 _Ralf Stephan_, Jan 21 2005