A101477 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.

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, 1584, 1, 2, 8, 40, 223, 1219, 4909, 9152, 1, 2, 8, 40, 224, 1327, 7371, 29506, 54912, 1, 2, 8, 40, 224, 1343, 8250, 46099, 183043, 339456, 1, 2, 8, 40, 224, 1344, 8427, 52938, 295915, 1164387, 2149888

Offset: 0

Author

Ralf Stephan, Jan 21 2005

Keywords

• nonn
• tabl

Examples

			1, 1, 3, 12,  56,  288, 1584,  9152,  54912,  339456, ...
1, 2, 7, 31, 156,  851, 4909, 29506, 183043, 1164387, ...
1, 2, 8, 39, 211, 1219, 7371, 46099, 295915, 1939395, ...
1, 2, 8, 40, 223, 1327, 8250, 52938, 347941, 2330532, ...
1, 2, 8, 40, 224, 1343, 8427, 54625, 362833, 2456261, ...
1, 2, 8, 40, 224, 1344, 8447, 54887, 365688, 2484384, ...
1, 2, 8, 40, 224, 1344, 8448, 54911, 366051, 2488831, ...
1, 2, 8, 40, 224, 1344, 8448, 54912, 366079, 2489311, ...
1, 2, 8, 40, 224, 1344, 8448, 54912, 366080, 2489343, ...
1, 2, 8, 40, 224, 1344, 8448, 54912, 366080, 2489344, ...
		

Links

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

Crossrefs

Rows converge to A052701. First row is A000257.

Cf. A052701, A101478.

Programs

Mathematica

nmax = 11;
b[x_] = Sum[2^(n - 1)*(2*n - 2)!/(n - 1)!/n! x^n, {n, 1, nmax}];
c[x_] = 0; Do[c[x_] = x*(1 + c[x])^4/(1 + c[x]^2) + O[x]^nmax, {nmax}];
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[n_, k_] := SeriesCoefficient[a[n, t], {t, 0, k}];
Table[T[n - k, k], {n, 1, nmax}, {k, 1, 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+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.