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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

Table of n, a(n) for n=0..65.

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.

Cf. A052701, A101478.

Sequence in context: A199915 A209557 A183759 * A077887 A059379 A065487

Adjacent sequences:  A101474 A101475 A101476 * A101478 A101479 A101480

KEYWORD

nonn,tabl

AUTHOR

Ralf Stephan, Jan 21 2005

STATUS

approved

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Last modified September 24 05:29 EDT 2021. Contains 347623 sequences. (Running on oeis4.)