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

Author

N. J. A. Sloane

Keywords

• nonn
• tabf
• nice
• easy

Examples

			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, ...
		

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

Crossrefs

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

Programs

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 *)