A101486 - OEIS

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

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OFFSET

0,3

COMMENTS

Rows converge to A005159.

First row is A000168.

LINKS

Table of n, a(n) for n=0..54.

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+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.

EXAMPLE

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,

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 *)

CROSSREFS

Cf. A000168, A005159, A101487.

Sequence in context: A057300 A076655 A236438 * A086606 A076112 A208744

Adjacent sequences: A101483 A101484 A101485 * A101487 A101488 A101489

KEYWORD

nonn,tabl

AUTHOR

Ralf Stephan, Jan 21 2005

STATUS

approved