A248748 Number of rooted binary trees with n leaves and each internal vertex colored in one of two colors.

1, 2, 4, 16, 48, 192, 704, 3072, 12032, 52736, 219136, 985088, 4218880, 19144704, 84066304, 387088384, 1725497344, 7989886976, 36128948224, 168658206720, 770103574528, 3611291549696, 16636941697024, 78453223194624, 363787840389120, 1721209150504960

Offset: 1

Author

Stanislav Sykora, Oct 13 2014

Keywords

• nonn

Comments

For n>1, a(n) is the number of bipolar networks one can build from n identical impedances by combining smaller networks either in series or in parallel.

Also for n>1, given two symmetric binary operations f(x,y) and g(x,y), such as two different means of x and y, one can use them (and just them) to form up to a(n) distinct expressions with n arguments x1,x2,...,x5.

Examples

			a(5)=48 because there are three binary trees with 5 leaves, namely, (1,((1,1),(1,1))); (1,(1,(1,(1,1)))); ((1,1),(1,(1,1))); and each of their four (5-1) internal vertices can be colored in 2 ways, giving rise to 3*2^4 = 48 possibilities. The "coloring" can be indicated by means of two different kinds of parentheses, for example (1,[(1,1),[1,1]]).
It also implies that 5 identical impedances can be wired together in 48 ways, iterating only simple series/parallel bondings.
Also, given two different means f(x,y) and g(x,y) of two numbers (e.g., an arithmetic and a geometric one), these can be combined to form 48 distinct means of 5 arguments x1,x2,x3,x4,x5. One such mean, for example, is f(x1,g(f(x2,x3),g(x4,x5))), corresponding to (1,[(1,1),[1,1]]).
		

Links

• Stanislav Sykora, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000992.

Programs

PARI

v=vector(1000); v[1]=1; \\ Use any desired size
for(n=2,#v, v[n]=sum(k=1,n\2,v[k]*v[n-k])); \\ v = A000992
for(n=1,#v, v[n]*=2^(n-1)); v \\ Final multiplication and result display

Formula

a(n) = A000992(n)*2^(n-1).