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Search: id:A119627
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| A119627 |
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Number of labeled graphs with no isolated nodes of size up to n+1 nodes, using 2*n+2 unique labels (no two nodes can have the same label). |
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+0
1
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| 6, 95, 3122, 202671, 25992373, 6561168159, 3271778102626, 3238332198581151, 6386927543425690577, 25167828012974622494207, 198457647877828107872246829, 3134149754118486012018252515615
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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Replacing 2*n+2 in the formula with m gives the forumula to calculate the number of labeled graphs with no isolated nodes of size up to n+1 nodes, using m unique labels (with m > n). An alternative (and much more complicated!) way to find the sequence is with the following recurrence (n>1, m>n): a(n,m)=a(n-1,m)+binomial(m,n)*(-a(n-1,n+1)+sum_{k=1..n*(n+1)/2}binomial(n*(n+1)/2,k)) a(2,m)=binomial(m,2)+binomial(m,3)+binomial(3,2)*binomial(m,3)
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FORMULA
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a(n)=sum_{k=1..n+1}binomial(2*n+2,k)*A006129(k)
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EXAMPLE
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a(1)=6 because with 1+1 nodes and 2*1+2 labels, you can
construct the following graphs:
1-2
1-3
1-4
2-3
2-4
3-4
We have 6 different graphs.
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MAPLE
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A006125:=(n)->2^(n*(n-1)/2); A006129:=(n)->sum('binomial(n, i)*(-1)^i*A006125(n-i)', i=0..n); A:=(n)->sum('binomial(2*n+2, i)*A006129(i)', i=1..n+1);
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CROSSREFS
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Cf. A006125, A006129.
Sequence in context: A078103 A014378 A058465 this_sequence A116158 A116289 A116256
Adjacent sequences: A119624 A119625 A119626 this_sequence A119628 A119629 A119630
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KEYWORD
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easy,nonn
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AUTHOR
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Delorme C. (delorme(AT)lri.fr), Lopez R. (lopez(AT)lri.fr), Soguet D. (soguet(AT)lri.fr) Jun 08 2006
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