OFFSET
1,2
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.
LINKS
Stanislav Sykora, Table of n, a(n) for n = 1..1000
FORMULA
a(n) = A000992(n)*2^(n-1).
EXAMPLE
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,(1,1))); 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]]).
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Stanislav Sykora, Oct 13 2014
STATUS
approved