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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, 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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,5
LINKS
M. Bousquet-Mélou, Limit laws for embedded trees, arXiv:math/0501266 [math.CO], 2005.
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.
EXAMPLE
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, ...
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 *)
CROSSREFS
Rows converge to A052701. First row is A000257.
Sequence in context: A199915 A209557 A183759 * A077887 A059379 A065487
KEYWORD
nonn,tabl
AUTHOR
Ralf Stephan, Jan 21 2005
STATUS
approved

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Last modified March 19 01:34 EDT 2024. Contains 370952 sequences. (Running on oeis4.)