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A248748
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Number of rooted binary trees with n leaves and each internal vertex colored in one of two colors.
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2
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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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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FORMULA
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EXAMPLE
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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]]).
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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