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.

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, 1111, 7838, 24057, 1, 3, 18, 135, 1133, 9833, 69275, 208494, 1, 3, 18, 135, 1134, 10176, 90959, 635279, 1876446, 1, 3, 18, 135, 1134, 10205, 95635, 868827, 5994584, 222646205

Offset: 0

Author

Ralf Stephan, Jan 21 2005

Keywords

• nonn
• tabl

Comments

Rows converge to A005159.

First row is A000168.

Examples

			1,2,9,54,378,2916,24057,208494,1876446,17399772,
1,3,17,119,932,7838,69275,635279,5994584,57872666,
1,3,18,134,1111,9833,90959,868827,8504314,84866778,
1,3,18,135,1133,10176,95635,928442,9236144,93646430,
1,3,18,135,1134,10205,96191,937361,9365984,95427597,
1,3,18,135,1134,10206,96227,938179,9381050,95673739,
1,3,18,135,1134,10206,96228,938222,9382179,95697199,
1,3,18,135,1134,10206,96228,938223,9382229,95698688,
		

Links

• M. Bousquet-Mélou, Limit laws for embedded trees, arXiv:math/0501266 [math.CO], 2005.

Crossrefs

Cf. A000168, A005159, A101487.

Programs

Mathematica

nmax = 9;
b[t_] = 2/(1 + Sqrt[1 - 12t]) + O[t]^(nmax+1);
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]&;
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[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 *)

Formula

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.