A036602 Triangle of coefficients of generating function of binary rooted trees of height at most n.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 2, 3, 5, 6, 8, 8, 9, 7, 7, 4, 3, 1, 1, 1, 1, 1, 2, 3, 6, 10, 17, 25, 38, 52, 73, 93, 121, 143, 172, 187, 205, 202, 201, 177, 158, 123, 99, 66, 47, 26, 17, 7, 4, 1, 1, 1, 1, 1, 2, 3, 6, 11, 22, 39, 70, 118, 200, 324, 526

Offset

0,11

Links

Alois P. Heinz, Rows n = 0..12, flattened

E. M. Rains and N. J. A. Sloane, On Cayley's Enumeration of Alkanes (or 4-Valent Trees), J. Integer Sequences, Vol. 2 (1999), Article 99.1.1.

Index entries for sequences related to rooted trees

Example

Triangle begins:
1
1, 1;
1, 1, 1, 1;
1, 1, 1, 2, 2, 2,  1,  1;
1, 1, 1, 2, 3, 5,  6,  8,  8,  9,   7,   7,   4,    3,    1,    1;
1, 1, 1, 2, 3, 6, 10, 17, 25, 38,  52,  73,  93,  121,  143,  172,  187, ...
1, 1, 1, 2, 3, 6, 11, 22, 39, 70, 118, 200, 324,  526,  825, 1290, 1958, ...
1, 1, 1, 2, 3, 6, 11, 23, 45, 90, 171, 325, 598, 1097, 1972, 3531, 6225, ...

Maple

b:= proc(n, h) option remember; `if`(n<2, n, `if`(h<1, 0, `if`(n::odd, 0,
     (t-> t*(1-t)/2)(b(n/2, h-1)))+add(b(i, h-1)*b(n-i, h-1), i=1..n/2)))
    end:
A:= (n, k)-> b(k+1, n):
seq(seq(A(n, k), k=0..2^n-1), n=0..6);  # Alois P. Heinz, Sep 08 2017

Mathematica

b[n_, h_] := b[n, h] = If[n < 2, n, If[h < 1, 0, If[OddQ[n], 0, Function[t, t*(1-t)/2][b[n/2, h-1]]] + Sum[b[i, h-1]*b[n-i, h-1], {i, 1, n/2}]]];
A[n_, k_] := b[k+1, n];
Table[Table[A[n, k], {k, 0, 2^n-1}], {n, 0, 6}] // Flatten (* Jean-François Alcover, Feb 14 2021, after Alois P. Heinz *)

Crossrefs

Cf. A001190, A036587, A036588, A036589, A036590, A036591, A036592.

Sequence in context: A327104 A126061 A088496 * A176166 A167911 A037804

Adjacent sequences: A036599 A036600 A036601 * A036603 A036604 A036605

Keyword

nonn,tabf,nice,easy

Author

N. J. A. Sloane

Status

approved